It is important not to confuse correlation with causation, or causation with forecasting. A variable \(x\) may be useful for forecasting a variable \(y\), but that does not mean \(x\) is causing \(y\). It is possible that \(x\) is causing \(y\), but it may be that \(y\) is causing \(x\), or that the relationship between them is more complicated than simple causality.
For example, it is possible to model the number of drownings at a beach resort each month with the number of ice-creams sold in the same period. The model can give reasonable forecasts, not because ice-creams cause drownings, but because people eat more ice-creams on hot days when they are also more likely to go swimming. So the two variables (ice-cream sales and drownings) are correlated, but one is not causing the other. They are both caused by a third variable (temperature). This is an example of “confounding” — where an omitted variable causes changes in both the response variable and at least one predictor variable.
We describe a variable that is not included in our forecasting model as a confounder when it influences both the response variable and at least one predictor variable. Confounding makes it difficult to determine what variables are causing changes in other variables, but it does not necessarily make forecasting more difficult.
Similarly, it is possible to forecast if it will rain in the afternoon by observing the number of cyclists on the road in the morning. When there are fewer cyclists than usual, it is more likely to rain later in the day. The model can give reasonable forecasts, not because cyclists prevent rain, but because people are more likely to cycle when the published weather forecast is for a dry day. In this case, there is a causal relationship, but in the opposite direction to our forecasting model. The number of cyclists falls because there is rain forecast. That is, \(y\) (rainfall) is affecting \(x\) (cyclists).
It is important to understand that correlations are useful for forecasting, even when there is no causal relationship between the two variables, or when the causality runs in the opposite direction to the model, or when there is confounding.
However, often a better model is possible if a causal mechanism can be determined. A better model for drownings will probably include temperatures and visitor numbers and exclude ice-cream sales. A good forecasting model for rainfall will not include cyclists, but it will include atmospheric observations from the previous few days.
A closely related issue is multicollinearity, which occurs when similar information is provided by two or more of the predictor variables in a multiple regression.
It can occur when two predictors are highly correlated with each other (that is, they have a correlation coefficient close to +1 or -1). In this case, knowing the value of one of the variables tells you a lot about the value of the other variable. Hence, they are providing similar information. For example, foot size can be used to predict height, but including the size of both left and right feet in the same model is not going to make the forecasts any better, although it won’t make them worse either.
Multicollinearity can also occur when a linear combination of predictors is highly correlated with another linear combination of predictors. In this case, knowing the value of the first group of predictors tells you a lot about the value of the second group of predictors. Hence, they are providing similar information.
An example of this problem is the dummy variable trap discussed in Section 7.4. Suppose you have quarterly data and use four dummy variables, \(d_1\), \(d_2\), \(d_3\) and \(d_4\). Then \(d_4=1-d_1-d_2-d_3\), so there is perfect correlation between \(d_4\) and \(d_1+d_2+d_3\).
In the case of perfect correlation (i.e., a correlation of +1 or -1, such as in the dummy variable trap), it is not possible to estimate the regression model.
If there is high correlation (close to but not equal to +1 or -1), then the estimation of the regression coefficients is computationally difficult. In fact, some software (notably Microsoft Excel) may give highly inaccurate estimates of the coefficients. Most reputable statistical software will use algorithms to limit the effect of multicollinearity on the coefficient estimates, but you do need to be careful. The major software packages such as R, SPSS, SAS and Stata all use estimation algorithms to avoid the problem as much as possible.
When multicollinearity is present, the uncertainty associated with individual regression coefficients will be large. This is because they are difficult to estimate. Consequently, statistical tests (e.g., t-tests) on regression coefficients are unreliable. (In forecasting we are rarely interested in such tests.) Also, it will not be possible to make accurate statements about the contribution of each separate predictor to the forecast.
Forecasts will be unreliable if the values of the future predictors are outside the range of the historical values of the predictors. For example, suppose you have fitted a regression model with predictors \(x_1\) and \(x_2\) which are highly correlated with each other, and suppose that the values of \(x_1\) in the training data ranged between 0 and 100. Then forecasts based on \(x_1>100\) or \(x_1<0\) will be unreliable. It is always a little dangerous when future values of the predictors lie much outside the historical range, but it is especially problematic when multicollinearity is present.
Note that if you are using good statistical software, if you are not interested in the specific contributions of each predictor, and if the future values of your predictor variables are within their historical ranges, there is nothing to worry about — multicollinearity is not a problem except when there is perfect correlation.