 # Specification of AR Model

Do you think it will be important to know how to calculate autocorrelation? I understand the whole test and how to use the autocorrelation of residuals figures, but can’t quite get my head around how they are obtained. It seems these will normally be provided in a table.

Also, is it ok to conclude that there is no serial correlation based on the first few observations? On page 454 of the text book, there are 59 observations in total, and as I understand it, we would need to compare the t-statistics for each of the 59 autocorrelations of residuals with the critical t-value. However, only the first 4 observations are shown and they conclude based on that. Am I missing something?

I think they are assuming seasonality which can be discovered from 4 autocorrelations.

Even if they did give a whole page in the table on 454, you can eyeball the t stat for anything greater than 2 or so pretty quick.

Could someone also explain where autocorrelations come from. I get that it has something to do with the residuals at T and their lagged counterparts at T-1 (the occurence in the previous period).

If we get the correlation between these two sets of data, thats only 1 correlation figure. How do we get a correlation figure for each of the observations?

It is when the error terms are positivly or neg corr with each other. It is common in financial time series data and produces standard errors that are too small. Test it visualy or with DW and use 2(1-r) to determine pos or neg. For pos if the DW stat is below the lower DW band you have a problem. use Hanson method to correct

Follow differant steps if you are eval ARCH

Hi, I thought the point was that we need to use a t-test on the autocorrelation of residuals.

If it’s an AR time series, then you cannot use DW. It has to do with how in a time series, the next x depends on the previous x, which is why the data points seem to flow smoothly and not jump from one place to the other. So, if the current error is +5 points (i.e., the actual data point minus the point estimated by the model), the next error term will likely differ by a small amount (because next x is related to current x). That means you will see correlations between the erros, more severely than you would normally see in a linear regression. However, to make a judgement on whether that correlation is significant or not, you will have to check the t-stats. That’s my blurred recollection of this topic.

Yes, that’s exactly how I understand it as well. I’m just not clear on one of the inputs used in calculating the t-statistic, namely the autocorrelation coefficient.

It seems that its just the correlation between the residuals at t and residuals at t-1. On excel we would do =CORREL(A1:A11,A2,A12), we jump one cell because there is only one lag in an AR(1) model. This makes sense to me, but then my new question is, wouldn’t we need data far beyond the observations we are modelling? If we have 12 observations, the 12th autocorrelation coefficient would be =CORREL(A12:A22,A13:A23).

Isn’t it just the correlation between the residuals for an observation and a previous one?

To use the Durbin Watson test it’s the correlation between residuals for one observation and the one before it. Or (if you are looking for seasonality, for example ) it’s the correlation of the current residuals with the residuals at t-4 ( if it’s quarterly data).

You cannot compute the autocorrelations yourself…you will need to know the covariances and variances of the errors, and that’s why they give them to you.