A little help....I'm stuck... I've regressed my differenced series and believe that it has a MA part. I've backed into the MA(1) coefficient. So I have an ARMA (1,1) model with coeffecients. The next step is to test my model using a Box-Pierce Statistic or to test if the autocorellations of the residuals have a normal dist. The r's are calculated by eq 18.15 and the Q Stat follows, but the problem I'm having is coming up with the e^'s. How are these determined??? Thanks.
Jacob: How do we determine the residuals?
Rachel: The residuals are the actual interest rates minus the estimated interest rates. The actual interest rates are given. The estimated interest rates are based on the past interest rates and the ARIMA model.
Jacob: How do we test if the autocorrelations of the residuals have a normal distribution?
Rachel: Compute the sample autocorrelation function of the residuals and form a correlogram; we show an example in an illustrative spreadsheet. If the ARIMA model fits well, the residuals are a white noise process. Their standard deviation is 1//T, where T is the number of observations. Check the number of sample autocorrelations exceeding a given absolute value.
Jacob: If we have 400 observations, do we check 400 sample autocorrelations?
Rachel: The last 30 or 40 sample autocorrelations don’t have enough points. The first 4 or 5 sample autocorrelations can be distorted by other factors. Start with lags 6 through 55, for a total of 50 sample autocorrelations.
If only 2 or 3 sample autocorrelations are outside the 95% confidence interval, we presume the distribution is normal with the hypothesized standard deviation.
If 8 or 9 sample autocorrelations are outside the 95% confidence interval, we presume it is not a white noise process.
If 4 to 7 sample autocorrelations are outside the 95% confidence interval, we examine lags 56 to 105. We may have a white noise process with minor distortions. The ARIMA model may be reasonably good, even if it is not perfect.