# Time series - Number of observations

Let say we are doing time series analysis using 60 quarters of returns. (a) What is the number of observation T for AR(1)? (b) What is the number of observation T for AR(1) with annual seasonality? I found that in most CFAI textbook examples and EOC: (a) is 60 (b) is also 60 However in Schweser and one of the EOC: (a) is 59 (b) is 58 In some other textbooks, (a) is 59 (b) is 56 Has anyone noticed this inconsistency? I am so confused.

I did not do this from schweser. But, when calculating Standard Error for autocorrelations, which is 1 / Sqrt T, T is just the number of observations, without subtracting any degrees of freedoms. And, when determining critical t value from t table for autocorrelations, then degree of freedom is used as n-k-1. When in doubt, I guess we could rely on CFAI text.

i am confused as well and this is what i have found so far. lets say if you have 3 values from data about erm number of cars on road. and you want to use a one lagged function on it. that is c(t)= c(t-1) +Z then you can only plot 2 data points on the graph understand? these would be the third value against the second and the second against the first. And you have to find the regression line that best passes through this plots. SO u lose one observation. this is my uderstanding of it. and scweser has does it as well if you look the example there were 40 observations n he took 39. secondly what i dont understand how many will u lose if it is a function such as xt= x(t-1) + x(t-4). Here it says that T is basically the initial observations in the data minus the order of regressive function. So if there were 50 observations and it is a AR(1) function. then T is 49. But in the example it also says that a function such as xt= x(t-1) + x(t-4). is Ar(1) function with a seasonal lag not a AR(2) function but it calculates T by subtracting 2 and not 1 from the initial observation. for this case got it? it may also be important to note that if the question directly tels u the observations u had and not the the total amount in he original data the u don’t need to do anything\

I just checked my Stalla guide. For purposes of calculating the standard error, rus1bus is right. “T” is just the number of observations.

Not sure what the whole fuss is all about, so let me explain it in terms of my simple mind. The number of observations is depending on how many data sets you put into the model. Say you have a series of 60 quarter returns. You want to predict future returns using past returns, so you want to use AR(n) model and want to know what time lags are most significant. You start slowly first with AR(1) model. Number of data sets (T) for this case is 59 and t-stat this case is 59-2= 57. You calculate autocorrelation of residuals of this model. The standard error is (1/sqrt(59)). You see that the t test suggests that residuals are serially correlated at time lag 4 as well. so now you modify your model to be AR(1) with seasonality = xt = a+ bx(t-1) +cx(t-4). UNLESS you have access to data further back, you are now down to 60-4= 56 data sets. You test again autocorrelation of residuals of this new model. T is 56 and df= 56-(2 variables -1)= 53 degrees of freedom. You are happy to see that the autocorrelation of residuals are now insignificant so you can start using this model for further testing. Before you can use this you need to test for heteroskedasiticity (ARCH) and unit root. How many data points do you have for this data serie: now it is 56 data (set points) so you can use to test. In exam, you are unlikely to be confused how many observations (data sets) are used since it will clearly be indicated. Hope it helps.

Yes elcfa, it helps great with your explanation. Thanks for sharing your insight.

elcfa ur explanation makes sense but schweser does not treat it that way. schweser is not consistent and apparently cfa text books tell another story. and stalla say don’t deduct anything as one guy pointed out. look at question 16 at the end of time series topic in schweser. schweser does not deduct anything. for ar(1) model. look at page 233 of schweser it deducts only 2 observations for R(1) with seasonality = xt = a+ bx(t-1) +cx(t-4).

floyd Sorry. Don’t use nor have access to Schweser so cannot comment on that. Be glad to comment if you can post the whole specific text/example so that I can understand their specific comment. CFAI is consistent with my explanation, as far as I know. Ref treatment of sample size in foot notes on pages 432 and example 6 from page 410 Let me know if you have questions about them.

ok here an eoc from scheweser a monthly time series of changes in payroll expenses at a major retail chain was regressed with an AR(1) from January 2001 to June 2009 (102 months) the result of the regression and the first 12 lagged residual correlations are show in the next two tables. for calculation t stat in lag 1 shchweser uses t as 102

Right. Again since I don’t have more info about how Schweser runs the regression, I need to put out some conjecture. If Jan 2001 is regressed based on DEC 2000, Feb 2001 is regressed based on Jan 2001, you have 102 data sets (observations) then t = 102. Since they use t= 102, I must guess that is what they did. This is the same method CFAI uses in example 15 (see footnote 34) If Schweser does not use DEC 2000 --> you have to start from Feb 2001 is regressed based on Jan 2001 --> 101 data sets --> t = 101. This is the same method CFAI uses in example 4 for Intel sales. Note that the observations comes from 60 (in example 2 and 3) down to 59 in example 4. A final note (or two) I. As the CFAI text says, this calculation is based on methods developed by Diebold(2004). I know that there are several other methods of calculating this standard error of autocorrelation (ACF is the buzz word). SPSS alone gives you TWO different methods. SPSS calculations are much more common. Box Jenkins formula is the standard in industry. http://www1.uni-hamburg.de/RRZ/Software/SPSS/Algorith.115/acf_pacf.pdf II. You are not going get wrong answer if you miss it by 1 or 2 degrees. In exam, you will get clear instruction (I believe) what observation numbers to use (if the exam is THAT detailed). It has been normal that you are given all the key numbers served (thus a torrent of numbers), so the critical skill is to find out what numbers to pick (then do a simple calculation) to reach your conclusion. So relax. U r doing fine. Let me know if there are more questions.

Just double check the LOS about the requirement, just to make I have not misspoken myself. It says “explain how autocorrelations of the residuals can be used to test whether the autoregressive model fits the time series” Action word: “explain” has much lower requirement than ‘calculate’ and consistent what I had explained about expectation. Just thought you want to know.

firstly thank you for your detailed answer, the question does not give further info on what was regressed against what. after this there is just a table of the estimated coefficients and autocorrelations. Though i sort of get you mean. Its basically getting confused in the language or words. If you know you have to use 102. Then you can read the question statement in a manner which totally makes sense. But based just on this info I would understand if someone takes one observation out cos he does not know whether this is the original data or after the regression. As you said it is probably not that important and maybe its just a badly phrased question. Thanks again

btw where did you get all those terms ssp and jenkins box, are you a phd student or what!!! lol