Interpreting diagnostics and tests with time series data

Question

I am analysing the effect of monetary policy on output and inflation during crisis and after. Monetary policy is represented by exchange rate, interest rate, money supply and indicator of systemic risk + dummy financial crisis.

I´ve chosen multiple regression analyses with time series to assess differences. The response variable, being a time series, has autocorrelation and heterogenity. I tried to change functional forms in different combination, I used logarithm and finally first differences for all responses. However, after performing these transformations, statistical tests still indicated that autocorrelation was present in these data.

My questions:

1. In model with the first differences, the Durbin-Watson test obtains the test statistic 2.6. I interpret this as do not reject the null hypothesis, these data are consistent with a lack of autocorrelation. Is this interpretation right? Please reference the image . I worry that the correlogram residuals looks strange. I don´t understand what this means. Please check the second image. Do somebody know what does that mean?

2. If I use Breusch-Godfrey test or Ljung Box test of autocorrelation with 20 lags, results show, that there is not autocorrelation, but when I use only 5 lags, the Ljung Box p-value is under 0.05 and we reject the null hypothesis at the 0.05 level and conclude there is autocorrelation. Are these results contradicting one another? How do I choose which set of results to report?

Answer 1

Not sure if the answer is still useful for you after all these months have passed, but here goes:

@1: In your first regression there is evidence of negative first-order autocorrelation, as indicated by "rho", by the Durbin-Watson stat, and by the first bars in the correlograms (which repeat the rho value, but give you confidence intervals as well). So no, I don't think your suggested interpretation is right.

I'm not entirely sure what you mean with "looks strange", the different signs of the autocorrelations at different lags are certainly allowed and possible. But you're right that simple processes such as an AR(1) wouldn't look like that. There could be MA terms in the residuals, but the most important thing is that you are estimating an equation without dynamics (only contemporaneous terms), and it is not too surprising that there is still serial correlation left.

@2: At face value the different test results indeed contradict each other. However, it is obvious that the autocorrelation at lags 1,2,3,5 is relevant. Your test up to lag 20 is mixing in a lot of insignificant partial effects and thus destroys the power of the test. This is an example of the old mantra that a non-rejection shouldn't be treated as an acceptance of the null hypothesis. It is true that in general it is not clear or easy which lag length should be tested. But in this case I think the case is pretty clearcut, namely that there is a problem with remaining autocorrelation.