This is one of the most important questions one can ask about the residual time series that results from having fit a model. If we can say "Yes" then we have a useful indicator that our analysis may have gone as deeply as it can go.
One article that I hope everyone will review at least briefly is the classic,
The Ljung-Box test is used in many hundreds of papers, far more than actually cite the source of the test. Oddly enough, the papers that speak directly to the test, rather than simply apply it, tend to be critical. In fact, we will often observe in our own explorations that the Box-Ljung test has a hard time rejecting the null hypothesis. This lack of power has been explored in several papers, most notably in
This brief well-written paper provides a useful paradigm for the investigation of model misspecifications. Besides the natural estimates of the power of the Ljung-Box test, the paper also considers the "percentage increase in expected squared forecast error one step ahead which results from using the misspecified model." This interesting quantity helps one decide if the misspecifications matters or not. This is important since defenders of a misspecified model will often argue that the alleged misspecifications really doesn't hurt. Sometimes this is true, but often it is bologna.
Another, more recent, follow-up on the Ljung-Box test considers time series with heteroskedasticity or with simple nonlinearities.
There are many articles by Box that are relevant to our course. A couple of interest that don't quite fit here, but this is were they will live for the next few weeks.
Box, G.E.P. (1966) "Use and Abuse of Regression," Technometrics, 8 (4) 625--629.
This article describes in a simple way one of the basic problems of applying regression to observational (as opposed to planned experimental) data. The main problem is that of "latent variables," or those explanatory elements that we were unable to include in our model. There may be earlier sources for the basic ideas here, but the exposition is still well worth reading.
We will look at several articles on the comparison of forecast accuracy. This is a easy one to help start the conversation.
Brillinger, D.(2002). Teaching Note on ARIMA Fitting of the DJIA.
This elementary exposition suggests how as an exercise one might go about fitting an ARIMA model to the price series of the Dow Jones Average. It motivates differencing on an intuitive basis and does model selection by minimization of the AIC. In our class, we will so similar exercises at various levels of sophistication.
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