Thank you.
Good morning everybody and welcome to the lecture on time series econometrics in the MSC program economics.
This lecture is as you all know, due to the pandemic again a lecture which takes place in digital format, only.
There is a recording which is already on, unlike the previous semester the recording will always start right at the beginning of the lecture.
And for interaction our possibilities are limited to to data protection reasons of which you're probably aware so we cannot talk to each other in the normal way, but there is the possibility of posing questions to me via the chat function.
There is in fact not only the chat function in zoom but also a function called question and answers, and I would like to ask you to use the chat function throughout the lectures, only the chat function proposing questions.
This makes it a little easier for me to see what kind of questions are being asked, rather than having two channels which basically serve the same purpose.
So please use the chat function if you have a function I typically see on my screen when the question comes in, without being able to see the question itself so I will then typically, at some point,
break and look at the question and then answer it or answer a couple of questions which may have come in, and I hope I can give you satisfactory answers.
The lecture is three hours per week.
On Mondays, I will typically make a break of say five minutes or something like this just to have a coffee or cup of tea and after one and a half hours approximately and then we will continue with another 45 minutes of lectures.
Unfortunately, the lecture will be a little shorter than usual for a semester, because there are two holidays on Mondays in this semester.
One was Easter Monday last week so we didn't have a lecture there and the other one is Pentecost Monday, where there will also be no lectures so there are 12 weeks of lectures, rather than 14 weeks of lectures.
This is relevant on the one hand for the amount of material I'm able to present in the lecture and on the other hand for the student contributions, which are due typically each week, starting with the second lecture.
For today's lecture, no student contribution is due and for last lecture of the semester, the 12th lecture, also no student contribution is due and actually you cannot submit a student contribution there, which leaves you with the possibility of submitting 10 student contributions.
You have to submit seven student contributions in order to be admitted to the final exam, which is an oral exam in both the first and the second slot.
So the procedure is as it was in my lecture on advanced metrics estimation inference. The last semester I suppose many of you attended this lecture.
One week after a lecture, you have to submit a one page full of handwritten notes which summarize the content of the previous lecture you have to submit this digitally and send it to an email address which will be provided to you by my secretary.
The precise regulations on the student contributions, you can find on a steener where I uploaded file organization and syllabus and student contributions so informing you about all these things.
Please have a look at this file and familiarize yourself with the conditions for for the student contributions again if you have attended my lecture on estimation inference last semester you will find most of that are quite familiar.
The student contributions are a means for you to review and repeat what I have done in lecture and to identify what you think is important for you to know.
And you should do this carefully because you have the possibility of taking your handwritten notes, those of your notes which we accepted a student contribution with you for the oral exam.
So it is allowed to make use of them during the oral exam when I ask you a question and you tell me you would like to have a look at your notes that's fine and I give you some time for that.
And you may find the right answer there, which I think should be an incentive for you to make good use of the student contributions and properly such that they will be helpful you in the process of learning and in the process of examination.
And are there any questions from your side that please raise your hand to let me know.
You can then type in the question in the chat function.
I do not see any indication of questions. So let me start with the lecture whose topic is as the title says time series.
Time series econometrics and what I will do today is that I start with a simple introduction to introduce you to the concept of time series, as opposed to the type of data structures we've been dealing with in the lecture of the previous semester.
So, what is a time series, very simply, a time series is a series of observations on a certain variable equally spaced in time.
It is of course important that there is some kind of structural constancy over time for time series. So, in particular, it is important that the definition of the variable does not change over time, which is a problem which we frequently encounter in time series analysis.
That time series are discontinued at some point in time or continued with a differently defined variable.
Some of these differences in definitions may be of minor importance or we may hope and assume that there are minor importance, some of the differences may however also be very important.
Like, for instance, when you look at German GDP, you will see that we have a breakthrough break there in the middle of the 1990s due to German reunification.
Since East Germany joined the former West Germany, then of course there's a major structural break in time series. Actually, it is not then anymore a series of observations on the same variable but the variable is differently defined.
I have supplied some example data sets with German and United States data.
I start here in this introduction with time series on German data, of which you find two data sets with the extension WF1. That's the extension for the commercial software eViews, which we are using in this lecture.
I did not yet mention this software, but you find the information on that in my remarks on the organization of the lecture. You can download a student version of eViews free at a link which I supplied in this organizational document and which was also sent to you by my secretary.
I hope most of you or all of you have already installed eViews. Unfortunately, the student version of eViews is not a full version of the limitation which the manufacturer has implemented on the student version is quite a severe one.
Namely, you are not able to save your eViews work files. So while you can work with an open work file just the way typically eViews users can work with it, you can never save your results.
And if you want to preserve your results, therefore, please think about printing them into say a PDF file or printing them out physically or something like that, which of course limits the access then to the results on later occasions because you cannot load the PDF files in eViews, but at least you have saved and documented the results.
This limitation, I hope, will not last for long because I have already a long time ago applied for a full eViews campus license of the university.
And I have been informed that the university is in the process of acquiring this license. So I hope it won't take much longer anymore. It does take already very long for the university to acquire this license, but I'm confident they will do it.
And then you will receive access to a full license of eViews and store the data sets. But for the time being, it is actually not really a great problem because currently you have data sets supplied, work files, eViews work files supplied, which you can open with a student version and then you can look at the data and do some simple transformations to reproduce what I show you in this lecture.
There's nothing of such a big importance that you really need to modify the eViews work files in this lecture or for what we do in the next lecture. And afterwards, I hope that the full version of eViews will be available.
Now, in this data set, either quarterly data or monthly data, so Q for quarterly and for monthly, you find a number of variables and I take three of these variables now to show you certain properties of time series.
One is the variable GDPW, which is real GDP per worker, which I take from the data set with a quarterly frequency, and then I look at the short run nominal interest rate in the data set with monthly frequency and at the rate of return on the German stock market index DAX,
which is also in a monthly frequency or I have transformed the data such that they are obviously they're usually daily frequency, but I have transformed them or picked them just on a monthly frequency to make things easier.
Since daily frequency for data is already something which causes trouble typically because there are some days where observations are not recorded, let's say holidays, for instance, on financial markets, then there is a problem with the definition of the time series
already because we require from time series that the data are equally spaced in time. And for some of the financial variables with daily frequency, there is actually no equal spacing in time, maybe perhaps not for holidays, maybe not for weekend data or something like this.
And so this does not really correspond to exactly to the definition of the time series, unless we think of the time series as being equally spaced in time but having certain missing observations, which then causes other problems which we would like not to encounter in the introduction to time series.
So it is better to use a monthly frequency where the data set is complete.
Here is the first graph which I show you which is the real GDP per burka in Germany from 1991 so this is already post unification up to the post recent observation which is available.
So last quarter, fourth quarter of 2020. And what you see here is a time series which is essentially upward trending with lots of peaks and troughs.
And some of the troughs stand out, in particular, this truck here, and this truck here and it is obviously not difficult to identify the cause of these trucks.
This year is the severe recession in 2008 2009 of the financial crisis, and this is of course, the corona crisis.
So, that's easy to explain this phenomenon we are often interested in say when we do business cycle analysis to look at recessions and also up swings which follow a recession, which hopefully also follow the corona crisis here.
But what we see in this time series apart from the general upward trend of the time series is also a sequence of peaks and troughs at rather high frequency.
So you can see that this year is basically an annual phenomenon which we see here.
We have very regularly spaced ups and downs in this time series. And this is of course what we call the seasonal effect of time series.
So the within year fluctuation of the corona data.
Typically, the peak in this time series here is the last quarter. So the quarter in which Christmas season boosts sales by companies and typically the traffic is then the first quarter of the new year, we have much less sales take place and then it's going up again.
So that is what you often do not see in economic time series anymore these type of fluctuations, because very often we work with series seasonally adjusted data.
One point of the introduction which I give today to you is that I will talk about seasonal adjustment and also about problems which are associated with seasonal adjustment about the way we alter the data which
we have measured already prior to our analysis by applying seasonal adjustment techniques in a data series. So this is why I start here with an example of time series which has not yet been seasonally adjusted.
And one interesting feature is actually that the variations which we have in the process of the seasonal happenings in the time series are very sizable.
They are not quite as sizable as the major recessions we see in the financial crisis in the corona crisis, but they typically match the amplitude of fluctuations which we have in less severe recessions over the course of the business cycle.
So, you don't see any kind of business cycle fluctuation actually between 1991 and 19 and 2008 in this time series, despite of the fact that for instance in 1993 we had a major recession in Germany.
This is more or less hidden behind the seasonal variation. And you also don't really see the rather say depressive date of the German economy in the early 2000s in this time series, because the seasonal variation dominates this type of information in the time series.
So this is one reason why people prefer to look at seasonally adjusted time series. And as I say, we will discuss seasonal investment in the course of this lecture.
Now here's the second time series which I announced. This is the short run nominal interest rates for three months money in Germany, this time over a very long horizon monthly data from the end of 1959 so December, 1959 to March, 2021.
And it just shows you what kind of interest rates short run nominal interest rates have been recorded in Frankfurt and actually this is already a time series which is not exactly of the same definition of during the length of this period here, because for the first part of the sample.
This is really three months interest rates recorded at the banking place in Frankfurt, and then this series was discontinued and I use the oil reboot three month interest rate.
Continue it. You don't see the point where the change takes place and actually data are very similar, but strictly speaking, it's not the same series at the end.
Not the same series with which we start at the beginning of the series.
Now, this series looks very different from the one we have seen previously. First, there's certainly not an upward trend in the series and second, it is not really clear whether there is perhaps downward trend in the series.
Some people think that interest rates are on a secular decline and then they point out this development which we see since not so clear perhaps since the beginning of the 1990s, perhaps a little later that interest rates seem to go down over time.
Whereas interest rates, nominal interest rates have stayed rather constant, not really constant, but they didn't really have a clear trend between 1960 and say 1995.
There were certainly great variations in interest rates during this period of time of some 35 years, but very clearly there is no trend, no clearly discernible trend in the first 35 years of the sample.
And it may be that the pattern has changed since, or at least we have a very pronounced downward movement here where it's not really clear whether it is completely a trend or what else is happening here.
I mean, from an economic point of view, it is not really very likely that this trend will continue because we have already arrived at negative interest rates here and very clearly interest rates cannot become arbitrarily negative because then people would prefer to hold just currency rather than save anything.
So it actually cannot be really a long run trend which we see here, but we do see a very pronounced lowering of the level of interest rates over a considerable period of time of now, say 20, 25 years or something like this.
So the trending behavior of this series here is very difficult to assess from a graphical perspective and it's actually also difficult to assess it from an econometric point of view.
The second thing which is interesting in the interest series is that while this is not original data which I show you here, there is no seasonal effect visible in this time series with perhaps some exceptions here at the beginning of the sample.
Indeed, formal tests would show that these peaks here are seasonal peaks, but later on nothing like this is visible anymore. So for all practical purposes which we have today, we typically do assume that there are no seasonal effects in interest rates, but the time series shows you that this is not naturally the case and that may actually be the case that interest rates are not visible.
We do see in this case that interest rates also respond to seasonal fluctuations in the rest of the economy as we have seen this in the early 60s, but we haven't seen it since.
And by and large, we do not really see seasonal effects in this series here.
We do see pronounced up swings and down swings in the interest rate. So this is some type of cyclical activity, perhaps, which is reflected in these up swings and down swings and interesting economic question would be whether they correspond to our business cycles,
which we measure in, say, GDP or other main macroeconomic aggregates. So there's quite a good point to make that this is business cycle variants, which we see are the results of business cycle activities in the economy, which drive interest rates here.
And unlike the previous graph, which I showed you on Rio GDP per verga, the business cycles are very easy to discern here, whereas we did not have this easy identification of business cycle activity in GDP.
All right, so that is the second series I wanted to show you. And here I show you a third series, which is the rate of return on the market.
Now, the rate of return is actually something of which we should think that it is very closely related to an interest rate, because people who have financial resources at hand can either invest in fixed interest rate securities or they can invest in the stock market.
And they should typically choose the type of investment which gives the higher rate of return. So the short run nominal interest rate for length of three months should actually be economically, from economic theory, it should be closely related to the return on the DEX by means of, let's say, arbitrage.
But we don't really see this in the data here.
The rate of return on the DEX, which should be closely related to the rate of return on three months securities, fixed interest rate securities, is very, very different from the previous time series.
What we see here is lots and lots of ups and downs. Very clearly, these are not seasonal ups and downs. You see this monthly data again for the same time period which I showed you for the normal interest rate.
The ups and downs are much more frequent than we would expect for monthly, for seasonal monthly effects, and they seem to be rather irregular.
So it seems to be the case that there is lots of noise in this data here. And one challenge we have in time series analysis is to extract the interesting systematic information from a series which is very noisy, like this series on the rate of return on the DEX.
We will later come back to this series here and see that there are actually rather easy techniques on how we can smooth this series in a certain way in order to see more clearly if there is any interesting systematic information in this series.
At first sight here, you would find it probably difficult to see anything systematically here. It looks almost like a random process.
All right. These are the main differences which I wanted to emphasize. To summarize it, you find it again here on this slide. So GDP per worker has a trend.
R does not have a trend, so the rate of return on the DEX does not have a trend. I didn't mention this just when I showed you the time series, but it was pretty obvious.
Whether or not the normal interest rate I has a trend is unclear, as I said.
GDP per worker has a seasonal component, whereas the two rates of return normal interest rate and return on the DEX do not have a seasonal component.
The normal interest rate has a cyclical component. This is at least unclear for GDP per worker and certainly unclear for the real rate of return on the DEX.
And then the latter series contains lots of noise, as we have seen much more so than GDP per worker at the nominal interest rates.
Noise is what we typically use for shocks which have no discernible structure. We will give this a precise definition in the lecture.
What I do here is that I introduce already four different concepts of phenomena which we see in time series.
And I will shortly make the point that we can think of a time series of being composed of, let's say, the sum of these different concepts.
So a time series may have a trend, it may have a seasonal component, it may have a cyclical component, and it may have noise to various extents.
So in some time series, the trend may dominate, and other time series, the noise may be dominant, and third time series, the seasonal component may dominate.
So if we think of a time series as being made up for such phenomena, then we must also think about how important are these phenomena or what kind of weights do trend, season, cycle and noise have in a particular time series.
So in some sense, time series is a problem of signal extraction. We have, let's say, a sum of four different phenomena, and we are interested in perhaps just one of those components, perhaps just in the cyclical component.
So we need to extract the information on the cycle from the information which perhaps we are not so much interested in like the trend and the season and the noise and that's the called signal extraction problem.
And there are even more problems, of course, associated with time series and one of them is, and that is why I start out here with German data, that some time series may contain structural breaks.
So, the prime example for that is German GDP before and after German reunification, and I show you this time series now I denote it GDP SA, which is the real seasonally adjusted.
This is what SA stands for seasonally adjusted, quarterly GDP of Germany. So this is not per worker anymore I do not normalize GDP but number of workers, which of course also increased during reunification and there were for some extent takes into account.
So, in fact, that more GDP is created in the reunified Germany than in the pre unification West Germany.
But here I do not standardize GDP by dividing through the number of workers but I just look at the real seasonally adjusted quarterly GDP.
This is not the seasonal fluctuations or actually the statistical agency has done so. And now I just look at the quarterly values for GDP after seasonal adjustment for Germany between 1960 and this case it is 2007, where I stopped so well after reunification.
So this I think is quite a nice example for showing the importance of looking at data prior to analyzing data, because German reunification is not really something which is an eye catcher.
So if you if you look at this data series here and have no knowledge about German recent history, then you may actually overlook the fact that here is a jump in the series.
We may think this is perhaps just some kind of boom which Germany experience at some point in time and you think it is something interesting economically.
And while the German reunification also poses a lot of interesting economic problems.
The jump, which we see here is obviously not an economic phenomenon but political phenomenon, which result in the unification of Germany, and this caused the German population to increase by about 16 million people.
East Germany had 60 million East Germany had about 16 million people but people in East Germany were much less productive in terms of GDP than people in West Germany.
So, the jump is not as big as the number, the numbers of the population would suggest, but still knowing that this happened in 1991 so reunification was in 1990 but the 1990 data is still for West Germany and the 1991 data is for the unified Germany.
So knowing that this happened, we now see very well that here is a structure break in the series so the definition of the time series is not constant over time, and we need to take this into account when we work with data over this horizon here.
You see the same series for the full sample, where we have also the financial crisis in the time series and where we have the corona crisis in the time series, and then becomes even more apparent that there is a danger to overlook structural
crisis, because now this upswing here does not seem to be very normal compared to the upswing after the financial crisis, or the upswing which we briefly had during the first wave or after the first wave of the corona crisis.
So, an uninformed observer may not note at all that there is a structure break in this series.
But if he is to note that, then only by visual inspection. Typically, just using the data with say econometrics or spreadsheet software, which you have on your computer would take this as normal economic activity and would probably not warn you that something unusual
happened here. So, you should always closely inspect the time series graphically before you use it in any kind of econometric procedure to identify unnormal properties of the time series.
There may be structural breaks, there may also be outliers, data points which have not been recorded properly, or other exogenous events which perhaps distort the results which you are interested in for reasons which you could possibly correct if you have been warned of this phenomenon time.
Okay, so that's the first warning I have to issue in terms of data analysis. Many time series have irregular or unexpected features.
There may be structural breaks, there may be trend reversals, there may be extreme values, outliers, jumps or anything like this. And these type of events can cause really bizarre results in regression analysis.
Often, such features are easily detected by just plotting the time series prior to analysis. So please make sure that you always plot your series before you use them in more sophisticated analysis.
Always first have a look at the time series graphically at all the time series you want to look at. First have a look at it graphically and check whether something irregular or surprising has happened during the period which you are using in your analysis.
If you see something like this, then investigate what the reason of that may be and what kind of measures you may take, whether it is warranted to do anything about it. Sometimes you should not do anything about it, but this depends on the cause of the problem.
This leads me to warning two. Sometimes people see problems in the time series after they have looked at the time series. For instance, they see certain outliers.
And then they will take out the outlier or do something which is some kind of manual editing of time series, like, for instance, not only take out the outlier, but trim the time series by taking out, for instance,
10% of the observations and reducing the highest 10% and reducing the value to some other value which seems more plausible or something like this.
This kind of manual editing of a time series is very dangerous and actually it is usually not accepted unless you have very good reasons to do that.
So while it is easy to identify problems in the time series, it is not easy to do something about it.
If you were to manually edit the time series beyond, say, obvious mistyping errors which sometimes take place in time series,
if you do something about it which you cannot justify well, then people would suspect that discretionary changes to the time series bias your result and that you may have used that in order to find the results that you want to show in your paper.
So the appropriate approach to bizarre features of your time series is actually that you think about why these kind of irregularities exist and then try to find in your regression certain variables which can explain these irregularities rather than try to eliminate the irregularities from your dataset.
On the other hand, of course, if there is a typo or some other obvious error, then you can correct that and document it.
If you're sure it's a typo and then just say at this point has been shifted one or two positions to the left or to the left or to the right, then obviously you can do this and just for other researchers that the data has this feature and that this is the cure which is appropriate.
Now, as I have already pointed out, we can think of time series, assuming now that they have no structural break, that they consist of a long run component, which we call a trend, which I hear denote YT trend,
a medium run component, which I often call a cycle and I denote it here YTC for cycle, a seasonal component, which I denote YTS.
Seasonal component would mean that there are fluctuations with annual frequency for data which have higher frequency than annual, so quarterly, monthly, weekly, daily, hourly or something like this, all this may have seasonal frequency.
Obviously, annual data will not have a seasonal component in annual data and then there may be unstructured noise, which I call YTM.
That's a standard view of how a time series may be made up of different components.
Actually, there is no necessity for having just these four components. For instance, there may also be short cycles with less than annual frequency, which would then not be seasonal cycles,
which would also not be captured by a medium run component here. So you could think of other components, six components in such a composition of a time series, but the standard view is to think of these four components here.
One way to look at that would be to say that the time series, which we observe, which is YT, so for the observation we make in period T, that this is an additive time series or that the composition is additive.
So we just add up the trend composition, the cycle composition, the seasonal composition and the noise composition.
But obviously, this need not be the case. We may have time series where the components of the time series are indeed trend cycle season and noise,
but they are not made up of some of these, or the time series is not made up of some of these four components, but there's some other mathematical operation which ties the four components to each other.
One easy alternative is for its multiplicative composition of those four components, but even that is not necessarily the case. You can think of many other ways to combine trend cycle season and noise with each other and need not be regular, even in the sense that some of the operations are multiplicative or all of them are multiplicative.
There can also be some multiplicative and some can be exponential and some can be just added or whatever.
Usually, economists look for linear structures like here in equation one, but a linear structure typically already implies that there is a multiplicative structure like in two,
in some other transformation of the data. So for instance, if we analyze data in logs, as we usually do, and we assume that the logs have this additive composition of the four components, then clearly the original
levels, which I denote capital letter C of y, so small y is the log of capital Y, the original data, not in logs, would have the multiplicative structure if it had be added.
Now, what time series economists are interested in, at least when they work with applied interest, then this is the decomposition of the time series into its trend of typical seasonal and noise component.
And I will now talk first a little bit about trend elimination. Actually, I will talk a lot about trend elimination during the course of the lecture because there are different possibilities for trends that can be linear, deterministic trends, they can be nonlinear trends, they can be
stochastic trends to which we come at the end of the lecture. And therefore, there are also different possibilities to eliminate the trend. And unfortunately, one particular way of trend elimination is not necessarily appropriate for all possibilities of trends with this data. So we also need to know what kind of trend model
explains the data well before we do the trend removal if we are not interested in the trend. Clearly, sometimes in time series analysis, the trend is exactly what we are interested in, for instance, when we talk about economic growth and we are interested in the trend.
But for other questions, for instance, for business cycle analysis, at least traditionally people have thought that it is easier to look at business cycles when we remove the trend from the data.
And later on, other people came along and say, well, that's actually not innocuous to do this and we may change the cyclic properties of the time series when you eliminate the trend in a certain way. These are problems we will talk about later in the lecture.
But here's one very easy trend elimination technique. I go back in this case to the example of a GDP per burger. And what I have done here is just that I compute the first difference of this time series GDP per burger.
So taking the first difference means, let's just look at the change in the time series. And since a trend, if it is a linear trend, has always the same change from one period to the other, we would not see the trend anymore. We would just see a constant increment each period.
So we would see the trend suddenly as a constant in the time series and not as a trend anymore. This is precisely what happens with GDP per burger here when I look at the first difference. First difference means I take the difference between period t and period t minus one.
We will, for other purposes, sometimes also look at other differences, in particular for quarterly data, we will often look at the fourth difference. So between period t and period t minus four. So just the same quarter in the previous year, that's also a reasonable way to transform time series if it is of quarterly frequency.
But what I do here is that I just look at the first difference. And now you see a time series which has no trend anymore. It still has the seasonal variation very clearly in the data, we can still see two troughs which are pronounced, particularly pronounced actually the financial crisis here and the COVID-19 crisis here.
We also see a very pronounced recovery after the first wave of COVID-19 in the German data right at the end of our sample here. And as you see, it's going down again. This is when the second wave comes. But we don't talk so much about COVID-19 now. And then we see the seasonal variation in this time series. So that's one possibility to remove a trend.
The interesting question is, however, is the cyclical information in this time series and also the seasonal information and perhaps even noise information, is this still the same after we have removed the trend by first difference?
So that is not that clear. Is it possible, this is not the question, that removing the trend by building the first difference, here we're denoted by the delta operator, that removing the trend by building the first difference, alters in particular, perhaps the cyclical information.
Suppose we are not so interested in seasonal information and noise components, but we may be really interested in the cycle from a business cycle perspective. Then we have to make sure that removing the trend does not alter the cyclical information.
Now, suppose that delta GDPW has still exactly the same cyclical information.
Well, if this is so, and if it is true that the time series is additively composed of trend, cycle, season and noise, then we would know that trend component is, well, I denoted here, the trend component of GDPW would be the original series minus the first difference.
Minus the first difference of the series, because the first difference of the series would then contain cycle, season and noise, and if cycle, season and noise are not changed, then we would know that the difference between the original series and cycle, season and noise, which is embodied in the first difference here, that this would be the trend component.
Okay, so if it is true that first differencing does not change cycle, season and noise, then we would be able to identify the trend component as the difference between the original series and its first difference.
Now, what would this mean? This would mean the trend component is equal to GDPW in period t minus the first difference of GDPW in period t, which is GDPW t minus this difference here.
I mean, GDPW in period t and minus the same variable in period t minus one, then you see the GDPW here and there canceled and we are left with GDPW in t minus one.
So this shows us there's something wrong with this trend adjustment because we would have computed here that the trend component is just equal to the lagged series, lagged by one quarter.
Why does this tell us that this is probably not the right identification of the trend?
Well, we know the original series GDPW has a seasonal component and therefore GDPW just lagged by one period also has a seasonal component.
So we do know that GDPW, lagged by one period, has a seasonal component, so it cannot be just a trend component.
And the same argument we can make for cycle and for noise. Obviously, since GDPW t has a cycle, GDPW t minus one as a series also has a cycle.
And both of them also contain noise. So very clearly, going back to this page here, the lagged value of the series cannot be the trend series.
Or put it differently, if we think that this lagged value here is the trend part of the series, then we would know within in this trend part here, we also have cyclical information, seasonal information and noise information.
And therefore we must have altered cycle, season and noise in the first difference here. So the first different season, the first different series does not give us the same information on cycle, season and noise as the original series has done.
And therefore, apparently, this method of trend removal, which seemed to be very successful because there was no trend in the series anymore, is not actually the appropriate way of eliminating trend.
Unless, I must say, there is a very specific trend model to which we come later, namely that this is a stochastic trend, that there is no cyclical information and so forth.
But in general, this year is not a good measure of the trend. And therefore, we haven't really removed the trend properly from the data, but we have removed more than just the trend.
We have also removed some part of the cyclical information, some part of the seasonal and some part of noise information. And therefore, the inference we can draw from the series different ones is different from the inference on cycles we would draw from the original series.
So bottom line is data GDPW has not the same cycle and also not the same season and noise component as the original series.
This is the warning. Number three, first differencing of data alters the time series information beyond mere trend elimination.
It alters the information in the series in a way which we have actually trouble to completely understand. So that's why I warn you of just taking the first difference of a series if you think you're not interested in the trend, because you change the data by first differencing
and you get different results when you analyze the first difference data with respect to say cyclical properties, then you would have them if you analyze the original data with respect to business cycle questions.
This warning applies both for variables and logs and variables in levels. So it doesn't make a difference here. It of course does presuppose that the original time series is additively made up of the four components, but the same argument would carry over to say multiplicative or other combinations of the four components.
So very clearly first differencing can get us the probability.
Moreover, first differencing also resides in the loss of information, because we cannot recover the first observation of GDPWt anymore if we just work with the data GDPWt series.
So think of this symbol here as denoting not only the information in a particular period t on the series, but think of this symbol here as denoting the complete information we've had over the time series for any t.
Say t going from period one to period capital T or something like this, then we cannot recover the first observation GDPW1 if we have just the series data GDPWt.
This information has been lost because it has been subtracted and can now be found in the series data GDPW.
Okay, so first differencing of the levels gives us of course the absolute changes.
Often, researchers think that more informative are relative changes so the growth rates. So the growth rates we may denote by say g tilde or variable y period t, which would be y t minus y t minus one divided by y t minus one.
These would then be percentages in decimal notation. So, unlike absolute changes, the growth rates are independent of the units which the variable has been measured and that is one of the advantages of working with growth rates rather than working with absolute changes.
However, the growth rates such as g tilde are asymmetric in the following sense.
Suppose we have a very simple time series in which y t minus one takes on value 100. The next value y t is 125 and the next value y t plus one is 100 again.
So what we would see here is that the growth rate from period t to period t minus one is 25% right 125 over 100 minus one is 25 is 0.25 so it's 25%
However, the growth rate from period t plus one to period t where the absolute change is just the negative of the change we had here. This growth rate is just negative point two.
So we have 25% growth here and we have 20 negative 20% growth here.
It seems that growth rates are different or growth rates are actually different, even though we arrive at precisely the same value at which we started. So in this sense, growth rates as defined in equation three are asymmetric.
So what people do instead is often that they use the first differences of logs.
The first differences of logs give what is called the continuous growth rate and you note this not by tilde, but just by g y t where now I use a small y for the log of capital Y.
So I look at the log of y t minus the log of y t minus one, which is y t minus y t minus one in small letters or delta y t.
So that would be the continuous growth rate, the first difference of the logarithm of the series. And this is symmetric.
Look at this. Suppose y t minus one and y t plus one are hundred, just as in the example up here, and y t is 125. So same example as up here.
Then we would find that y t minus one and y t plus one have the same log, which is 4.605.
And y t is the log of capital Y t here, small y t is 4.828. So obviously the continuous growth rate g y t and the continuous growth rate g y t plus one are just the negative of each other.
So it's 0.223 here and negative 0.223 here. So in this setting, we have the same absolute value of the growth rates corresponding to the same change in the underlying original time series, which is kind of a nice
property and justification for why economists typically work with the difference of logs rather than with absolute changes or with exact discrete growth rates, as I have shown it here.
You're probably certainly aware that the log of a variable can be approximated as the variable minus one. So the log of x is approximately equal to x minus one if x is approximately one.
So it is true that this continuous growth rate g y t as the difference of the logarithm, so the difference of log of y t minus log of y t minus one.
That, well, this is the log of the ratio of y t minus y t minus one. And supposing that y t and y, that this ratio here is close to one, which is my previous example wasn't exactly actually quite very close to one because it was
a growth of 25%, which is a lot much more than we usually see in quarterly or yearly data.
But supposing that this ratio is close to one, we can then apply this approximation here is approximately equal to y over y t minus one minus one.
And that is the same thing as y t minus y t minus one. So the change in the original variable divided by the previous value of the original variable, which is g tilde y t.
So as long as the growth is not too strong, g y t is approximately equal to g tilde y t. And even for growth of 25%, as we had it here, 25% here or minus 20% here, well, it's approximately equal.
22.3% here or the negative of this here is somewhere in between these two ratios and not too far, actually, given the higher rate of growth.
All right. And so this is another reason why people like to work in first differences, even though as we have seen first differences.
First differencing is not innocuous, even though first differencing may lead to changing will lead to changing the cyclical information in the data.
There are other methods of trend elimination and no method is accepted throughout. All of them have their particular problems and some of those methods and the associated problems we will discuss in this link.
So the next exercise for you here, would you please solve at home. Take the German data set, the quarterly data set in Eeveeux and compute both g tilde and g for the seasonally adjusted variable on real GDP, and then just compare the results graphically.
And then convince yourself that it doesn't really make a big difference, whether you take the discrete growth rates or the continuous growth rates, so that given its advantages, it is well certainly advisable to use continuous growth rates rather than discrete ones.
The rest of the lecture I will actually spend on the seasonal component, because I would like to warn you on the dangers associated or the phenomena associated with seasonal adjustment.
And we will start here in just the same way in which we have just thought of removing the trend component from the data.
Because if you have a data series which is seasonal like our original series GDP W, then you may think well one way to eliminate the season from the data is to compute the fourth difference of the series, which I hear the
write down as data GDP W for to indicate that is the fourth difference. Actually, the four should perhaps better be moved to the data operator here to make sure it's the fourth difference which we compute on GDP W.
But now I have defined it this way here, so this thing is GDP W t minus GDP W t minus four so minus the value recorded in the same quarter of the previous year, rather than the value, the previous quarter, as we had done it in our first approach where we try to eliminate the trend from the series.
So doing this I arrive at the red time series in this graph here.
And I have also shown you the original data in blue. You see the original data has the very clear seasonal pattern with the ups and downs in annual frequency.
And you now see in the red data that we don't have such a regular component anymore. There are still ups and downs in this series, but we cannot say anymore that there is a clear seasonal pattern in the time series rather what we certainly have in the time series is some type of noise.
And certainly there's also some type of cyclical information in the data. For instance, if you look at this information here, then you see that over a number of quarters GP was tricky.
And that is what we call the 1993 recession, when over one, two, basically three, four, five quarters, a German GP was going down or at least the growth rate was going down.
The growth rate becomes negative only here, but the growth rate was declining and we had a recession in German GP.
So some cyclical information is in here and probably also in here. This is noise in here and the trend is also going away. So we have apparently removed not only the seasonal component or some of the seasonal component.
We have also removed the trend. The series here does not have a trend anymore.
We can and must however pose the same question as before. Does GDP W4, how this series, which we defined as the fourth difference of the original data, have the same cyclical and noise components as the original data.
Suppose that this is the case. Suppose that the cyclical and the noise component has not been changed by fourth difference.
If this were true, and if the composition is additive of the four components of GDP W and the original data, then it should be the case that the difference between GDP WT and data GDP W4T should just consist of trend and season.
So if the cycle and the noise have not been changed, and if the information on cycle of noise are now completely contained in data GDP W4, this variable here, then it should be the case that the original series GDP WT
minus the series which contains all the information on cycle and on noise
is a series which contains just trend and season.
Now let's look what this difference here is. This is GDP WT minus the difference between GDP WT and the same series lag by four periods.
So GDP WT cancels, we are left with the series lag by four quarters.
By the same argument that I have already made when we eliminated the trend by first differencing, this series here of course also has a cycle and noise component.
So does not consist only of trend and season. So very clearly the data GDP W4 series here has different information on cycle and noise than the original series.
By this rather primitive way of removing the seasonal information from the data, we have also altered the cyclical information on the data.
And that's a problem because this biases our results. We do not work with the original data anymore. We have removed some information on particular in particular the cycle from the data.
So we work on incomplete and biased distorted information when we work with data GDP W4.
Yeah, so forth differencing eliminates more than just trend and season and therefore alter substantially the information in the series with respect to cycle and noise.
There's another problem with this type of seasonal adjustment because the fourth differences remove the seasonal component of the series only if the seasonal pattern doesn't change over time.
The seasonal component is the same in each fourth component of the quarterly series.
If this is the case that the season that the seasonal effect is constant in each quarter.
So the third quarter of this year is the same seasonal effect as in the third quarter of the previous year, then obviously we do correctly remove the seasonal information for differencing.
But what we know from seasonal data from the analysis of seasonal data is actually that the seasonal effects are typically not constant over time.
So since these patterns change over time for differencing is general, not a reliable method of seasonal adjustment, and it is not used by statistical agencies, as I will show you later.
However, sometimes researchers use it as a shortcut to seasonal adjustment when they do not want to do a full adjustment for.
There are also legitimate reasons why people don't want to use the seasonal adjustment by statistical agencies, in particular because it's hard to understand what exactly happens to the data when statistical agencies apply their very complex methods of seasonal adjustment.
Sometimes people prefer to use this kind of crude removal of seasonal effects as done by the fourth difference filter, but it has its problems, as I have told you.
You may also ask, I mean, using the fourth difference filter removes the seasonal information from the data if the seasonal information is constant and otherwise it removes parts and perhaps important parts of the seasonal information from the data.
But it also eliminates the trend as we have seen.
So, fourth difference serves both purposes, removing the trend and removing much of the seasonal variation in the data.
And we may ask if it is possible to eliminate the season, also without eliminating the trend.
For instance, we are interested in the trend because we want to study growth or we want to study the interaction of growth and cycles, then we may be interested in not removing the trend information but removing the seasonal information which sort of
distorts or hides the information on trend and cycle which we find in the data.
Well, this is exactly what seasonal adjustment aims at to remove just the season and not also the trend.
One way to do this, also a crude way to do this is to compute what is called the smooth component of GDPW.
The smooth component of GDPW would be a weighted average of four successive quarters.
So, you build GDPWt plus GDPWt minus one plus the value in t minus two plus the value in t minus three. So you sum the four values of a year, not necessarily calendar a year but over time for each quarter you sum this value with the previous three quarterly values and divide the whole thing by four.
Then you have what I call the smooth component and then denoted here by GDPWs for smooth.
So that's basically what we call a moving average of the series which is provided by this definition here. It's an unweighted moving average because all the four components of the moving average receive the same weight which is 0.25.
Now this year's GDPWs typically has no seasonal component anymore and it has usually only little noise but the cyclic information is greatly altered.
The trend component is unaltered by the cyclical, by this unweighted moving average only if the trend is linear.
If the trend is not a linear trend then the trend component would also be altered.
So here you see the smooth component of GDP per worker in blue and the original series in red.
The seasonal fluctuations are easily discernible in the original data and there is no seasonal effect anymore in the blue data so that's why it's called the smooth component.
Otherwise, it follows quite nicely the general development of the red series here. In particular, we very clearly see the effect of the financial crisis and the success of upswing and we do see the effect of the COVID-19 crisis here.
However, there is change in the cyclical information which you can also see in this graph because you see that the downswing here occurs later in the smooth component then it occurred in the original data and correspondingly the upswing also occurs later.
That's a general feature of this type of seasonal adjustment, also a crude way of adjusting the season in the data that the cycle is shifted in time and therefore the cyclical information is altered.
Now as an exercise here, take the data which I provided to you, you find all of them on Steenen, on the nominal interest rate and on the real rate of return on the DAX and compute the smooth components for this series and compare them with the original series.
In particular, have a look at the R-series and its smooth component under the question of whether there is actually a cycle in this DAX returns. Recall the original series on DAX returns was very, very noisy and there was very little we could actually see that which seemed to be of interest.
Now compute the smooth component for these variables here and have a look at them and see whether you find something cyclical in the DAX series.
Alright, this was the introduction. Are there any questions so far?
Please raise your hand if you want to type in the question.
Okay, good. Then let's move to the first concepts and tools that we develop and here I talk about lag polynomials.
I'll start with the definition. The lag operator L is defined by a certain property, namely L applied to a time series yt or a single observation yt shifts the time series or shifts the observation by one period backwards in time.
So L applied to yt gives yt minus one and this is true for all t.
There's a certain mathematical structure behind the lag operator which is connected to the so called Z transform. This is advanced mathematics which I will not present in this lecture.
Unfortunately, there is no way of explaining the lag operator mathematically without the Z transform so I cannot really give you a rigorous definition of the lag operator here.
Rather, I do what you also actually find the most textbooks on time series analysis and just define the lag operator by its properties.
For those of you who know a little mathematics or operator theory, the lag operator is linear operator. So the usual rules on linear operations do apply to the lag operator.
We can also define an inverse operator, L-inversity. L-inversity applied to yt then shifts yt forward in time by one period. So it's yt plus one.
Why is this true? Well, very easy. Just multiply this equation here by L from the left hand side on both sides of the equation. So you would have L times L-inversity, which is then first cancelling, times yt is yt, is equal to L times yt plus one.
When L applied to yt plus one is of course yt, then you would find that yt is equal to yt. So this basically proves that there is an inverse operator and we denote the inverse operator by L-inversity.
As this notation with an exponent already suggests, we can use powers of the lag operator because repeated application of the lag operator makes sense and just results in shifting the variable further in time.
So applying the lag operator twice, we define this as L squared of yt. This is just L applied to L yt. Well, since L yt is yt minus one, this is the same thing as L applied to yt minus one. So this gives yt minus two.
So L squared times yt is yt minus two. The general rule is you just take the exponent of the lag operator and make this the number of lags applied to the time series. So generally we would have L to the power of tau times yt is the same thing as yt minus tau for all possible taus.
Obviously, if the lag operator refers to a constant, then we have L times nu, say the constant is nu. L times nu is equal to nu because nu is, in some sense, it's a time series which is constant in time, so it doesn't change.
So shifting this time series by one period doesn't change the value of the time series, so it stays new. So L times nu is equal to nu, a lag operator referring to a constant just disappears, the value stays constant.
From this fact that L times nu is nu, for any constant nu, it follows that the absolute value of L nu is the same thing as the absolute value of nu. Just taking absolute values of this equation here.
But the absolute value of L times nu is the same thing as the absolute value of L times the absolute value of nu. So we can, for any nu different from zero, divide this equation here by nu and arrive at the conclusion that the absolute value of L is equal to one.
So the lag operator has an absolute value of unity.
From this, it follows that the absolute value of nu times L is equal to one exactly if and only if nu is an absolute value equal to one.
And this property here that we now know what the absolute value of nu times L is or when this absolute value is equal to one, this property will be important when we invert lag polynomials because the inversion of lag polynomials,
the geometric series is used and the geometric series does not converge if the coefficient usually called C in the geometric series has absolute value one.
We will see that when applying this to time series and lag operators, we will need to know when the absolute value of nu times L is equal to one as the boundary case where the geometric series doesn't converge anymore.
And now here we have derived already the conclusion that this thing here is equal to one precisely when nu has an absolute value of one.
Apart from the lag operator, there is also the so-called difference operator, which I defined in this green box here.
The difference operator is denoted by capital delta, so this triangle is a capital Greek letter delta, and it is very easily defined as delta is equal to one minus L.
So the first difference of the time series yt is delta yt, which would be one minus L times yt. So just multiply the yt into this one and minus L here. This gives you yt minus yt minus one, which is the first difference.
We use first differences many, many times in time series analysis. We also use this in economics often because some time series have the property that the first difference is perhaps even more interesting than the level of the time series.
For instance, if small pt is the log of the price level of some economy, then the first difference pi t, usually denoted pi t here defined as delta pt is the inflation rate.
Sometimes people are interested in the way the inflation rate changes. So the change of the inflation rate delta pi t is delta pi t.
So it is delta applied to delta pt. So again, we can apply the operator two times in a row or can denote this as delta square pt, which is the same thing as one minus L square pt.
And now it is the case that with a leg operator, you can just work like with nominal polynomials. So you can apply the usual rules to one minus L squared. Namely, this is one minus two L plus L squared applied to pt.
So this is pt minus two pt minus one plus pt minus two.
That's the change in the inflation rate. And it makes sense to see this is the change in the inflation rate because obviously pt minus two pt minus one plus pt minus two is the same thing as pt minus pt minus one minus pt minus one minus pt minus two.
This here is inflation in period t, inflation rate in period t, this here is inflation rate in period t minus one. So here it is. So it's the change in inflation rate.
As I already said, one minus L squared is a polynomial in the leg operator L. And this is something you have to get used to. We will use leg polynomials a lot in this lecture.
So we very often work with polynomials defined not in a variable as we usually do say x or y or something like this, but a polynomial in L in the leg operator. It's something which is not a real number and not a variable but a linear operator.
And we can apply to such polynomials the usual rules of manipulating polynomials.
So the definition here is that an expression alpha of L defined as one minus well the sum of certain powers of L weighted by a certain coefficients alpha i that this is called a leg polynomial or a linear filter.
So I normalized this leg polynomial such that the power zero L to the power of zero and that this has a coefficient of one right alpha of L always has a coefficient of one for no leg at all.
And, well, this is also a convention to have negative sign here could also be different, but we divide with negative sign from this is subtracted a weighted sum of powers of L and the weights are given by some coefficients alpha i which can be positive, negative or zero.
The sum here has P components and I allow P to go to infinity. So P is called the order of the leg polynomial and this can actually be an infinite order polynomial. There's no particular conceptual problem in using infinite order polynomials.
If you dislike the summation signs you can of course also write this in extensive form alpha of L defined this way is just one minus alpha one L minus alpha two L squared minus minus minus up to alpha P L to the power of P, supposing that P is finite or up to infinity.
Obviously if P is infinity. So if we have P equal to infinity, then this requires that the sum of the alpha i know the sequence, excuse me, the sequence of the alpha i coefficients here is absolutely summable.
So we would then need to have the property that the sum over the absolute values of the alpha i is smaller than infinity. Otherwise, the sum would not converge, as you may recall from your lectures on linear algebra.
If we have two polynomials in the leg operators say alpha of L and beta of L, then the usual commutative law holds for the multiplication of the two leg polynomials.
So it is true that alpha of L times beta of L is the same thing as beta of L times alpha of L, which is a convenient property. As I say, the rules for manipulating polynomials, which you are used to from your analysis lectures or from school education just apply in the same way also to polynomials in the leg operator.
As we have shown, the absolute value of the leg operator is equal to one. And this implies then that the geometric series in the case of, say,
an upper limit of infinity here, the geometric series C to the power of i, L to the power of i, converges, provided that C is an absolute value smaller than one, converges to 1 over 1 minus CL.
So, very often we do not need to deal with complex expressions like infinite sums in the leg operator, but we will be able to write them in this form here, 1 over 1 minus CL, which we call an inverse leg operator, or sorry, an inverse leg polynomial.
So 1 minus CL is a leg polynomial, and 1 over 1 minus CL is its inverse. It would actually cancel if we applied 1 over 1 minus CL to the leg polynomial 1 minus CL. So if we multiplied 1 over 1 minus CL as an alpha with a beta of 1 minus CL, then we would find that the product of the two is just one and the leg polynomial cancels.
The proof of this property here is easy. We just apply 1 minus CL, so a simple leg polynomial, to this explicit form here of the geometric sum, and show that this product of 1 minus CL times the sum here is equal to 1, because this then proves that this infinite sum here is the inverse of this leg operator.
So it must be equal to 1 over 1 minus CL. And that's very easy. Now, multiply this sum here by 1, then we just get the sum, and then multiply this sum here by C times L, then this raises the power here by 1, so we would not have to the power of I, but to the power of I plus 1 here, which I have written here.
But rather than having I plus 1 here and I plus 1 here, and have I start at 0, we can also have I starting at 1, and going all the way up to infinity. So it's the same thing as this sum here, where I starts at 0 minus the same sum where I starts at 1.
Well, the end is always infinity, so there's no change. So the difference between these two sums is just the value of C to the power of I, L to the power of I, if I is equal to 0, this is C to the power of 0, L to the power of 0, which is equal to 1, and that completes the proof.
Good. This makes the first part of the lecture for today. I will now discontinue the recording for five minutes break, and later on, after five minutes, we meet again for the last 45 minutes of this lecture.
Okay, I hope you are all back. Are there any questions so far? Please raise your hand if you want to pose a question.
Okay. So what we have shown is that not only lag operators and lag polynomials exist, but also inverse polynomials exist. For instance, in the form 1 minus CL, parentheses, inverse C.
So we can both multiply lag polynomials with each other, and also divide lag polynomials by each other. One lag polynomial we can divide under certain conditions by another and possibly different lag polynomial.
The lag polynomial is a polynomial in the lag operator, but in principle, it is just a familiar mathematical structure. It is sometimes useful to evaluate the lag polynomial, not only with the lag operator as its argument, but rather plug in real numbers in place of the lag operator.
So for instance, while a lag polynomial alpha of L may be defined, it is useful to sometimes look at alpha of 1, so replacing the argument of the lag operator, the L, just by real number 1.
We then see that alpha of 1 is just 1 minus the sum of all the alpha coefficients in the lag polynomial. So it's 1 minus this sum here, and this expression comes up in time series analysis very often, so it is important that you familiarize yourself with the idea
that we can evaluate the lag polynomial at the point L equal to 1. We know already the absolute value of L is equal to 1, and it is quite important to see that alpha of 1 is sort of a special case of this lag polynomial because the absolute value of L is equal to 1.
Trivially, of course, alpha of 0 is equal to 1, because in the definition of the lag polynomial, perhaps I go back here, where was it here? If we have alpha of 0, so replace the L by 0 here, then all these powers of L will become 0, and we are just left with 1.
All right, so alpha of 0 is always 1 by definition, and alpha of 1 is just 1 minus the sum of all the alpha coefficients.
Since we know that the absolute value of L is equal to 1, we must never divide by alpha of L. If it is the case, then alpha of 1 is equal to 0, because then we would essentially divide by 0 if this is illegal.
This means that we cannot define the inverse of 1 minus L, say of the first difference operator, and there are other lag polynomials with similar properties which cannot be inverted.
This here would be illegal, is actually not defined, because the absolute value of L is equal to 1, and therefore 1 minus this thing here gives rise to something which is not defined, which is similar to a division of 1 over 0.
I can't explain it in a more exact way here, because we don't have the mathematical foundation for the lag operator, but I think the intuition should be clear.
Think of the L of something which is similar to 1, and this difference of them being similar to 0, therefore it's not allowed to divide by this thing, which means that the operator 1 minus L, the lag polynomial 1 minus L, or the difference operator, does not have an inverse.
Many, most lag polynomials do have an inverse, but some which are actually particularly interesting do not.
Apart from this, most familiar algebraic operations can be used to manipulate the lag polynomials, so for instance, 1 minus L to the power of 4, which is the fourth difference of a time series, which we used in this lecture for eliminating trend and season from a time series.
This can be written as 1 minus L squared times 1 plus L squared, so the third binomial formula, which you know from 9th grade or something like this, so we can factorize lag polynomials into sometimes easier factors, in this case 1 minus L squared and 1 plus L squared.
And obviously, we would be able to factor 1 minus L squared into 1 minus L times 1 plus L using the same third binomial rule again.
If alpha is an absolute value different from 1, we can also simplify some expressions in the lag operator. For instance, if we are given a ratio, a division of lag polynomials of the form, for instance, 1 minus alpha squared L squared divided by 1 plus alpha L,
we can recognize that 1 minus alpha squared L squared is the same thing as 1 minus alpha L times 1 plus alpha L. This is divided by 1 plus alpha L, so the 1 plus alpha L cancels, and so this rather complicated looking ratio is the same thing as 1 minus alpha L.
One important thing you should remember is that the lag operator must never change the variable to which it refers. So, for instance, when we have yt minus 1 squared, so just the value of y in period t minus 1 squared, we can write this as L times yt because this is yt minus 1 times yt minus 1.
But this is not the same thing as yt times L times yt minus 2, so we cannot think of the lag operator as being some type of constant by which we multiply, which we can then
also put at another position within the product because yt times L yt minus 1 would be the same thing as yt times yt minus 2, and if you did this operation, which is illegal and not possible, then you would suddenly prove that the squared value of yt minus 1 is the same thing as the product of yt
times yt minus 2, which is, of course, usually not the case. So always make sure that the lag operator does not change the variable to which it refers.
Now, as I pointed already out, we used this lag polynomial 1 minus L to the power of 4 in the application of GDP per verga called the result delta GDPw4t, which is a rather primitive seasonal adjustment filter.
Look at this filter here because the rest of this lecture will deal with seasonal adjustment. 1 minus L to the power of 4 is the same thing as 1 minus L squared times 1 plus L squared, which is the same thing as 1 minus L times 1 plus L because this gives 1 minus L squared.
And the product of, or multiplied by the product of 1 minus iL, 1 plus iL, where i here is the imaginary unit, so the square root of negative 1, not the normal interest rate, right? So that's the square root of negative 1 here.
So we see that the lag polynomial 1 minus L to the power of 4 first involves the lag, the difference operator, 1 minus L. So we may write 1 minus L to the power of 4 times yt as 1 plus L times 1 minus iL times 1 plus iL times delta yt.
Because this 1 minus L here is the delta operator and by the commutative law, which I explained to you, we can shift sort of this 1 minus L here and apply it to the yt to make delta yt from this. So in this case, we would have here a different lag polynomial now applied to the first difference of yt.
Further, we see from this equation that this filter 1 minus L to the power of 4, lag polynomial is also often called a filter, a linear filter, that this lag polynomial here has four roots.
That's to say four values for which it would disappear, for which it would assume the value 0, right? If we call the lag polynomial beta of L, so if we call the lag polynomial 1 minus L to the power of 4 beta of L, then we see from this expression here that beta of 1 and beta of minus 1 and beta of i and beta of negative i are all equal to 0.
Or to show it graphically, this lag polynomial beta of L defined as 1 minus L to the power of 4 has four roots on the unit circle and these roots lie at the intersection with the real and the imaginary axis. So here's i, here's negative i, here's 1, and there's negative 1 if we
are in the space of all complex numbers with the real numbers on this line here and the imaginary part of a complex number on the y axis here.
This cycle here, which gives us all the numbers with whose modulus is 1, so whose absolute value is 1, is very important in terms of testing for unit roots, which we will do later in this chapter, and I would just like to point out here that this lag polynomial beta of L has four unit roots.
One, which is here, which we'll later see corresponds to a stochastic trend, and the other ones i and negative i and negative 1, which correspond to seasonal unit roots, so to seasonal patterns.
What we have already pointed out is that seasonal adjustment by a four quarter moving average, which would be the polynomial 1 plus L plus L squared plus L3 divided by 4 is now turned to way of removing the seasonal component.
And now we see that 1 minus L times this filter gives us the filter 1 minus L to the power of 4. So the first difference of the four quarter moving average filter, which we used for seasonal adjustment, is the same thing as the fourth difference filter, which we also used for seasonal adjustment.
So the two crude methods of seasonal adjustment, which I had illustrated earlier, are actually related by each other in the sense that this moving average filter here multiplied by the first difference filter of the first difference operator gives us the fourth difference operator.
Since the first difference operator removes the trend, we now see that this filter here removes just the season and not the trend component, as does the fourth difference filter, which, as we had seen, removes both the trend and the seasonal component.
We have also seen that all these filters affect and change the cyclical properties of the series. And while I have demonstrated this by means of examples, I have not really given you a rigorous demonstration of this problem, that seasonal adjustment changes the cyclical information in the time series.
Actually, I cannot do this in the course of this lecture because a deeper understanding of this phenomenon requires spectral analysis. And I cannot do spectral analysis in this time series lecture for time reasons. There are other and perhaps more important topics to cover.
But I would like to give you an impression of what spectral analysis allows us to conclude in terms of seasonal adjustment, and therefore I would spend the rest of this lecture on spectral methods, noting that what is following now up to section 1.2.2 of the lecture is not relevant for the final example.
So what I do now is just for your information, you need not prepare this for the final exam, but I think it is extremely useful actually for understanding some properties associated with time series, in particular with time series which have not yet been seasonally adjusted or with time series which have been particularly adjusted.
And therefore I would just like to present to you an intuitive understanding of what spectrum of a time series is and what kind of features can be seen using spectral analysis.
So first, what is the spectrum of a time series? The spectrum of a time series is decomposition of the variance of a time series. So just suppose you have a time series.
This time series has certain variants and the spectrum now uses a certain mathematical technique called Fourier transform, which decomposes the variance of a time series across certain frequencies.
And the frequencies are mapped or numbers in the interval from zero pi, because all this relates to trigonometric functions, to cosine functions which have a certain periodicity, as you know, two times pi.
There's some symmetry so it suffices to look at just this interval. All of this is not so very relevant currently for the intuitive understanding, just to make clear why the frequencies we look at are zero and pi, because we're interested in cycles.
Actually, we are interested in seeing which type of cycle of which length is how important for certain time series. So there may be short cycles, there may be long cycles, there may be business cycles, there may be seasonal cycles, there may be many other types of cycles of different length and all that can be seen in the spectrum of time series.
Now the frequency, the highest frequency lambda equal to pi amounts in the spectrum to the shortest observable cycle.
Now this depends on the periodicity of the time series. Suppose you have a quarterly time series as the one we have worked with for GDP per worker. If you have a quarterly time series, then the shortest observable cycle is a half year cycle.
Because a half year cycle you can see, well, one quarter up, one quarter down, one quarter up, one quarter down, that would be a half year cycle. This you can observe with quarterly data.
You could not observe a quarterly cycle with quarterly data because you just have the quarterly information so you would have constant value each time, but you could observe a quarterly data with monthly, quarterly cycle with monthly data, of course.
So there's a well-defined concept of the shortest observable cycle and the shortest observable cycle depends on the periodicity of the data.
As I said, with quarterly data, the shortest observable cycle is a half year cycle and the shortest observable cycle is depicted at frequency pi.
For monthly observations, frequency pi would correspond to a two-month cycle, which is the shortest observable cycle for monthly data.
So the frequency, which is perhaps a little obscure concept now to you, is actually easy to understand if you know that 2 pi divided by the frequency equals the length of the cycle. That is to say, it equals the number of periods per cycle.
So to give you an example, for quarterly data, the peak at frequency lambda equal to pi over 2 amounts to a yearly cycle because if we have a peak in the spectrum at pi over 2, then 2 pi divided by pi over 2 gives 4, and four quarters is a year.
So while we already made it clear that with four quarterly data, pi is a half year cycle, pi over 2, so a shorter frequency, amounts to a longer cycle, namely a yearly cycle.
And correspondingly, if we look at frequency lambda, which is just pi over 4, 1 to 5 times pi, this would amount to a two-year cycle because this gives us eight quarters and eight quarters is two years.
And a peak at frequency lambda equal to zero would amount to an infinitely long cycle, so to something which is sometimes called a long-run trend component.
So let me give you some examples here. Suppose that we have epsilon t as just unstructured noise. We will later call this white noise, but this doesn't play a role now.
Just for our current purpose, epsilon t is unstructured noise, has no systematics in it.
And then suppose that we have a quarterly time series, which is xt is equal to negative 0.4 times the previous quarter value xt minus 1, plus some disturbance, some unstructured noise epsilon t, right?
Such a time series where xt depends negatively on the previous quarter value would exhibit Bing-Dang behavior with two cycles per year.
I give you an example here of a realization of such a series for 32 quarters, so for eight years, right? So this is just a simulation I have done.
It's artificial data of a time series generated by this process here. And this process, well, we can see some type of Bing-Dang behavior here, right?
A low value, a high value, a low value, a high value, a high value, a low value, a high value. So this seems very much Bing-Dang-like.
But not completely so. Here we have a low value, a high value, and an even higher value.
That it is not completely Bing-Dang is the result of the noise, which we have added to the series in this case, right?
But nevertheless, just get a visual impression of the series here. It looks somewhat cyclical, right? There is a clear cycle actually in it that we also see that it is rather short run cycle.
Yet it is perhaps not so easy to see what type of periodicity the cycle has. So let's look now at the spectrum of this process here.
The spectrum is actually the theoretical spectrum, which we can just compute from this expression here without looking at the real data.
The spectrum or the spectral density function of the cycle would look as follows. So this green line here, where you have on the x-axis the frequency.
The frequency I show here is numbers from zero to one half, point five, multiplied by two pi, right? So actually, this is one half times two times pi. So this here is frequency pi.
And I told you, frequency pi corresponds to the shortest observable cycle, which with quarterly data is a half year cycle.
So the spectral density of this process shows us, we have a peak here at the shortest frequency, namely at the half year cycle.
So this spectral density shows us the time series, which we have here, or this process here, generates a process which has, not surprisingly, a half year cycle, right?
Obviously, there's also other type of information in there. There's also variance at other frequencies. For instance, this frequency here, point two five, would amount to a yearly cycle.
There's also some variance here at the yearly frequency, as you can see, because the whole area under the spectrum, recall, is the complete variance of the series.
But most of the variance of the series is explained by the maximum here, by the cycle of a length of half a year.
There is, as I say, for instance, also cycle of a year length in here, which is quite natural, because if we have a half year cycle, then obviously, every year, there's also some kind of periodicity in the data, right?
This is why there's also spectral mass at point two five at this frequency, but this is not a pronounced cycle, because we have similar spectral mass elsewhere on other frequencies.
So, the interpretation you would get from this sort of spectral density function is actually that the time series we have, or the process which generated this time series, is a process which is characterized by one, or by a half year cycle, because here is the peak of the spectrum.
Now let's look at a quarterly time series, which is generated by a slightly different process. In this case, the process is x t is negative point four times x t minus two plus white noise.
So this is the very similar process, actually, same coefficient, negative point four, and now x is referring to x t minus two, so but to the value two quarters ahead of the current quarter.
A realization of this process using actually the same random numbers, so again, artificial data with the same random numbers, epsilon three t, as in the previous example, gives us this type of process here.
Compare this visually with the process we had here. You see there are more peaks here than there are peaks here.
But perhaps it is not so very clear at first sight what sort of the main periodicity of the cycle is.
So let's look at the spectrum. The spectral density for this process would have this shape here. So we would have, in this case, a clear peak at frequency point two five.
And now recall the cycle which is associated with frequency point two five is two pi divided by point two five.
So this would amount to four quarters.
Four quarters per year. So, sorry, it would be two pi divided by point two five times two pi.
So the two pi would cancel, we would have one over a quarter, which is four. So we would have just four quarters or a year as the main frequency, as the main cycle length.
This is why underneath the x axis I write cycles per period. This tells me this frequency point two five tells me that in a quarter, we run through a quarter of a cycle,
which then means that we need four quarters to run through the complete cycle and which would then with quarter data, implicate and imply that cycle is of length one year.
This is what this spectrum tells us. We have a one year cycle here and not a half year cycle anymore as in the previous spectrum.
Okay, let's continue these examples. Here I show you a quarterly time series xt, which is point four times xt minus four plus epsilon t.
Note that I have changed the definition here a little bit. First, now xt refers exactly to the value in the previous quarter in the previous year, so the same quarter in the previous year.
I don't have negative point four anymore, but I take a positive value here. Why do I do this? Well, go back to this cycle here. This cycle would imply xt is negatively correlated to the value two quarters earlier.
But the value two quarters earlier is again negatively correlated with the value two quarters, even two quarters earlier than that.
So xt would be positively coordinated with the value four quarters ahead of four quarters before in time. So this is why I use a positive number here to get a similar cycle.
The typical realization of this process here looks like that. And this looks already almost like real world data may look like.
So that would be a realization of this process here. And now let's look at the spectrum and see what kind of cycles we would find or what length of that cycles we would find.
Well, in this case, we would find much spectral mass at the long run frequency, which is actually sort of the trend in the series.
We would find much spectral mass at the yearly frequency. So that is a one year cycle. And we would find much, much spectral mass at the half year frequency.
So what you have to see is that this peak here and this peak here is what we call the season, right? These are cycles which we would see as seasonal. A yearly cycle is a seasonal cycle.
And a half yearly cycle is also a seasonal cycle.
What economists are typically interested in is not the seasonal pattern of a time series. It's not the half year cycle. It is not the yearly cycle, which is in this data here.
But they are interested in the business cycle, for instance. And the business cycle should typically have a cycle longer than one year and less longer than infinity.
So business cycles should typically be about this range of the spectrum.
In this particular series, which we see here, there is very little evidence for business cycle activities. Actually, here's the minimum of the spectrum rather than the maximum or something like that.
In the other time series, it could be different and we could have maximum of the spectrum here and then we would say, well, there is some business cycle activity going on in this series.
From this series here, if you just looked at the data in the time domain, you would not really know whether there is a business cycle pattern in this data or not.
You may actually say, well, there seems to be some kind of long run development. It's going up, up, up, up, up here and down, down, down, down, down here. So this may actually be some type of business cycle in Ireland.
But the spectrum very clearly shows there is no peak here at the business cycle frequencies, not in this range. The peak is in the yearly cycle, which is seasonal and in the half yearly cycle, which we have here.
So what does the seasonal adjustment to? Seasonal adjustment tries to remove the seasonal cycles from the time series. So it tries to remove the peaks in the spectrum at the frequencies pi over 2 and pi.
So precisely where we saw the peaks here in the artificial data. It tries to remove the variation in the time series, which is at this frequency and the variation of the time series, which is at this frequency here.
So suppose we have now original data, which is generated by a process like xt is equal to rho times xt minus some k plus epsilon t.
So k is any number, can be 1, 2, 3, 4, whatever. We leave that open. Suppose we have some original data, which we know, for some reason we know that it is generated by this process.
Then we would know the series xt can be given a representation epsilon t divided by the lag polynomial 1 minus rho l to the power of k.
Because obviously this process here can be written as lag polynomial 1 minus rho times l to the power of k, the whole thing times xt is equal to epsilon t.
So the data series xt has this representation here in terms of an inverted lag polynomial and white noise and a noise term.
Now suppose that we want to do some seasonal adjustment to this time series here, because maybe there are seasonal effects in there.
And suppose that we use the fourth difference to remove the solid seasonal component. This would mean that we multiply xt by the lag polynomial 1 minus l to the power of 4 to get a seasonally adjusted version of xt, which I call xt seasonally adjusted as a for seasonally adjusted.
xt seasonally adjusted is 1 minus l to the power of 4 times xt, and xt, sorry, xt is 1 minus l to the power of 4 divided by 1 minus rho l to the power of k times epsilon t.
So this here is the seasonally adjusted series.
What we would like to know now is what kind of effect does the seasonal adjustment fade filter 1 minus l to the power of 4 have on the xt, and how does it order the spectrum of the series when it gives us the seasonally adjusted series xt.
In general, it is the case that the seasonally adjusted series xt as a will differ greatly from the spectrum of the non-adjusted series xt if we use this type of crude seasonal adjustment.
And what this type of seasonal adjustment does to the original series is easily seen by now looking at the spectrum of 1 minus l to the power of 4 epsilon t.
So basically neglecting this like polynomial here and just looking at the seasonal adjustment filter divided by epsilon t.
Because this is the spectral density of the seasonal adjustment we do of the 1 minus l to the power of 4.
You see it has no variance at all at the long run frequency, no variance at all at the yearly frequency, no variance at all at the half year long frequency.
And it has some spectral mass in between here and there.
What you should understand now is that seasonal adjustment basically means that we multiply this spectrum here with the spectrum of the original series, so for instance with this spectrum here, or with this spectrum here,
or with this spectrum there. This is what seasonal adjustment is doing. The multiplication of the spectrum of the seasonal adjustment filter with the spectrum of the original series.
So we would like to study this now for the three examples that I gave you. I had three processes, one, two, and three, where xt refers to xt minus one or xt minus two or xt minus four.
And I now show you how the spectrum of this process here is changed if we seasonally adjust this process by the filter one minus l to the power of four.
And the same thing for this process and for the same thing for this process.
In the graphs which are to follow, the solid green line is always the spectrum of the seasonally adjusted data. You can just compute it, actually. It's not an estimation thing, it's just computation.
This solid green line is the seasonally adjusted data and the broken blue line is the original spectrum.
So the spectrum is not the original data. It's the spectrum of the original data, which you see here.
For the first process, recall we had the original spectrum as this dotted blue line here, which had a peak at the long run frequency.
The scaling of the vertical axis is now different. This is why it looks a little different, but if you compare numbers, you see this is precisely the spectrum we have already seen.
By seasonally adjusting the data, the spectrum of this process here is converted to a spectrum of this green type here.
It looks completely different, as you can see.
You see the data is indeed seasonally adjusted because there is no seasonal cycle anymore. There's nothing at frequency half year cycle, nothing at frequency yellow cycle.
It's completely taken out of the data.
But by seasonally adjusting the data in this way, we induce the peak here at a frequency of, well, that happens to be 0.125 actually, which is a two-year cycle,
which people could mistake as some kind of business cycle.
This is some type of artificially generated business cycle-like cycle, which is introduced into the data by the seasonal adjustment procedure we have used.
What I would like to warm you off is that seasonal adjustment, at least if it is done by crude methods like fourth differencing, which we use here, induces cyclical properties into the data which have not been there in the original data.
So this cyclical property here, which we see in the seasonally adjusted series, was induced by seasonal adjustment. It is not an original feature of the data, as you can see by comparison with the original data spectrum here.
Take the spectrum for the second process, which was this dotted blue line here, and see what happens for seasonal adjustment.
Well, multiplying the original spectrum with the spectrum, it's called a transfer function, with the transfer function of the fourth difference filter gives us this green line as a spectrum now for this seasonally adjusted series.
No seasonal information anymore, of course not at the yearly frequency and not at the half-year long cycle frequency. This is taken out of the data by the fourth differencing.
So this peak in the original data vanishes, but we generate two new peaks which have not been there in the original data.
And this here is again a peak at a frequency which is higher than annual, so people could mistake this as a cyclical, say business cycle or inventory cycle type phenomenon.
But it is not. It's a phenomenon which is induced by the seasonal adjustment.
For the third process, we get this result here.
This was the process where we had the positive coefficient and then referring to Xt minus four. The original data is the dotted blue line,
which had a seasonal peak at yearly frequency and at half-year long frequency, and no peak in the business cycle range, so there was the minimum.
This is completely reverted actually in the seasonally adjusted series, which is constructed by applying the fourth difference filter to the original series.
Here we again find a two-year long cycle in the seasonally adjusted data at frequency 0.125.
And also we find another short run cycle, which is completely foreign actually to the original data, but we do suddenly find it in the seasonally adjusted data.
So that's a fundamental problem of seasonal adjustment. Seasonal adjustment is likely to over adjust at the seasonal frequencies and to spuriously induce peaks in the spectrum, which were not present in the spectrum of the original data.
Which means that seasonal adjustment may spuriously induce non-seasonal cyclicality in data seasons and data series, which were not at all cyclical at the outset.
This makes it particularly problematic to compare cyclical features of model generated data. Let's say you have a DSGE model, for instance, and you generate data in the DSGE model with real world data, because the data you generate from your DSGE model do not include seasonal effects.
But real original data typically have seasonal effects, at least if it is GDP or employment or investment or something like this, and you calibrate the model at a quarterly frequency.
So real original data do have seasonal effects. And now the problem is how can you compare data generated from a DSGE model, which doesn't have any season built in the DSGE model, with real original data?
Well, the seasonally adjusted data may have spurious cycles. So you may actually compare the true cycles in your DSGE model with spurious cycles in the real world data if you use seasonally adjusted data.
But you cannot really compare the DSGE generated data with the real original data if these are seasonal because you don't have seasonal effects in the DSGE model.
So the comparison is quite difficult and much more difficult than researchers typically believe.
I'll briefly show you the same phenomenon for the seasonal adjustment with the moving average filter which I had discussed.
So the spectrum of this moving average filter, divided by four, is the spectrum here.
So we have value of one at the long run frequency. At the long run frequency the original spectrum is not altered because this is multiplied by one.
But then at the business cycle frequencies, actually variance is taken out of the data.
And finally, the variance is completely taken out of the data at frequency yearly cycle and frequency half year long cycle and not much variance survives in between.
So any kind of cycle which is of a frequency not really seasonal but less than a year is also mostly annihilated by this filtering, which was why the smooth component was so smooth.
Applying this now to our four processes, we see that we have here the original data for the first process which had the half year cycle.
Here was the half year cycle. Nothing of this half year cycle survives. Not much of variance at the high frequencies survives elsewhere, not at the yearly frequency.
But we've suddenly find a long run trend component in this series, which was there before, but now it is the maximum of the spectrum before it was the minimum of the spectrum. So this is quite misleading.
For the second process, the same thing. We had the peak here at the seasonal frequency. So at a one year cycle, this is completely taken out of the seasonally adjusted data.
No peak here anymore. Not at the half year frequency either. Not much in between. But suddenly we find here a long run trend as the peak in the spectrum, as the maximum of the spectrum, which used to be actually the minimum of the spectrum.
And then for the third process, we very well actually approximate the true spectrum of the data for the very long frequencies or for the very long cycles. This range here.
But then the spectrum deviates quite strongly already in the higher business cycle frequencies and flattens out completely for the very high frequencies which have a cycle length shorter than a year.
Obviously what I have shown you here are seasonal adjustments which are very crude. Those which are used in practice are much, much more complicated.
Because people want to prevent these kinds of misjudgments which may be
created by the primitive filters which I have shown you here.
So the seasonal adjustment filters used in practice which are called census X11, X12, X13 is currently the filter used in the United States.
Berliner for foreign views by the statistics just goodness I'm to Thomas Siat's filter which was constructed by my colleague my role, which has formed widespread acceptance.
They also do not completely solve the problem. They saw that much better, but not completely as I would show you.
There's a paper by Bell and Montsell which provides an analysis of the spectral representations of the census X11 seasonal adjustment filter which has been used by the United States statistical agencies, which
shows you what kind of effects this seasonal adjustment filter used in practice has had on the data.
It is very good to actually at the
at the low frequencies. So if the long run cycles, then takes out much seasonal mass at the seasonal frequency of one year, but has some annoying
waves in here and then takes out much of the spectral mass at the half year frequency again but already much earlier than only at the half year frequency.
And not all census X11 series are just by this filters. Sometimes census X11 works this way, which actually increases variance in cycles of a length between one year and half a year.
Or has these funny types of irregularities, which it creates the data, which you see minor peaks here and there and there. So there are different types of census X11 filters and also X12, X13 filters, which may again distort our inference and cyclical properties of the data.
This is it for today. As I said, this part was not relevant for the oral exam, but it may nevertheless have been interesting to you. I hope so, at least next time we start with stationarity and ergoticity.
Are there any questions? I'm sorry, I have already explored at the time a little longer than provided for, but perhaps you do have questions, then please raise your hand.
And yes, of course, I will answer them. Yes, somebody use the Q&A, please use the chat in the future. Do you have any recommended readings? Yes, of course, I do. You find them in the syllabus of the lecture.
In particular, I would recommend the book by Woltus and Kirchgesner on these issues, where, however, I must say that all of the readings I have will not deal with seasonal adjustment at the same level of detail that I have now provided to you in this lecture.
But the basic problem is recommended is out late in the Woltus and Kirchgesner. Any other questions? Then thank you very much for your attention and goodbye until next week.