# Autocorrelated predictors in linear models

I need to predict the outcomes of a time-series variable $Y$ based on two time-series predictors $X1$ and $X2$. For simplicity I will only illustrate $X1$ in the rest of this question.

The correlation between the predictors and $Y$ at lag 0 is relatively high as illustrated by the cross-correlation function. However the lag 1 correlation is a lot lower and does not seem to have much prediction power:

> ccf(as.numeric(X1), as.numeric(Y), lag.max = 3, plot = FALSE, na.action = na.pass)

Autocorrelations of series 'X', by lag

-3     -2     -1      0      1      2      3
0.015  0.008 -0.001 -0.415 -0.018 -0.002 -0.002


Indeed when I regress $Y_{t+1}$ against $X1_{t}$, $X2_{t}$, $R^2$ is very low.

Now, looking at the $X1$ partial auto-correlation indicates that the predictor itself could be estimated using an AR model. The significance level for this series is around 0.01:

> pacf(X1, plot = FALSE, na.action = na.pass, lag.max = 15)

Partial autocorrelations of series 'X1', by lag

60   120   180   240   300   360   420   480   540   600   660   720   780   840   900
0.171 0.103 0.067 0.055 0.046 0.049 0.027 0.032 0.029 0.012 0.020 0.037 0.020 0.026 0.020


To resume, the correlation between $X1$ and $Y$ at lag 0 is high (but not for $X1_{t}, Y_{t+1}$) and $X1$ is significantly auto-correlated up to 10/15 periods.

What would be the best approach to predict $Y$ and why?

1. Estimate $X1_{t+1}$ using an AR(15) model, then use the estimated value as a predictor for $Y$.
2. Fit all $X1_{t-15, ..., t}$ against $Y$.
3. Use an exponential moving average of $X1$ ($\alpha = ~0.125$) as the predictor.
• Is your goal primarily prediction or some form of inference? – cardinal Apr 6 '12 at 19:08
• Prediction is the primary goal. – Robert Kubrick Apr 7 '12 at 0:29