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I'm new to statistics and I'm currently working on some exercises to identify serial correlation visually. This is from a time series exercise of Dollar-Pound Exchange Rate.

Original Data

After running a simple OLS regression in STATA, i used these commands for the residuals:

predict residuals, res
gen residuals_lag = L1.residuals
scatter residuals residuals_lag

Residuals vs Lagged Residuals

I'm not sure what conclusion to draw from the second graph. The examples of serial correlation in my text book has a clear pattern and shape, but the example of no serial correlation has a wider random distribution than my output.

My guess is that the graph suggests no serial correlation as the residuals look to be random around 0, with a few outliers. But what do you guys think? Any input is appreciated!

Thanks!

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  • $\begingroup$ Can you show us a plot of the original data as well? $\endgroup$
    – user2974951
    Commented Jan 28, 2019 at 13:15
  • $\begingroup$ @user2974951 Sure! Have updated the original post. The data is a time series of Dollar-Pound Exchange Rate. $\endgroup$
    – Woo Won
    Commented Jan 28, 2019 at 14:09

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Exchange rates are often assumed to be autocorrelated for obvious reasons. Today's exchange's rate is a very good forecast of tomorrow's. It's easy to see without any graphs.

The question is whether the differences are autocorrelated? The simplest assumption is that they're not. Looking at your residual vs lagged residual, I'd say there's no obvious pattern. If you remove a couple of point, which could be outliers, it's just circular cloud in a scatter plot.

Often, the series such as these are modeled with ornsein-uhlenbek type of process: $$dx_t=a(x_t-\mu_t)+\sigma dW_t$$ where $dW$ - is Brownian motion, and $a,\sigma$ - are constants, and $\mu_t$ - potentially time varying deterministic or exogenous mean. This type of process would produce the residual plots similar to yours. It has characteristics of the random walk process (Brownian motion) in short run, and mean reversion process in long run. So, if you take differences, they'll look completely random, yet the levels will be persistent.

Finally, the exchange rate markets are heavily traded. To find such a simple auto-correlation in residuals is quite unlikely.

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  • $\begingroup$ A related question: to judge whether the differences are autocorrelated, why would one look at residuals from this regression vs lagged residuals, as opposed to looking at a time series plot of the differenced series itself? Thx. $\endgroup$
    – rolando2
    Commented Jan 28, 2019 at 14:41
  • $\begingroup$ @rolando2, res vs lagged res is a quick way to spot an order one autocorrelation. the residuals series will show autocorrelation only if it's very strong $\endgroup$
    – Aksakal
    Commented Jan 28, 2019 at 15:12

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