Can you explain the solution 2, especially the part of relevance of forward variable in regards to this line – “The same conclusion holds concerning the forward relationship. If the contemporaneous variable were defined so that it is realized at the same time as the variable we want to predict, the forward but not the contemporaneous variable would be relevant.”
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The language is a bit difficult to understand.
The following is what I can make out of the write-up:
When performing regression analysis we have a dependent variable X and an independent variable Y. The regression equation may be something like X = a + bY.
Now, if we are looking for the contemporaneous relationship between the two variables, we are essentially using the current value of Y to predict the current value of X. In the example given, the current value of bond yield (which is our independent variable, Y) is available at the beginning of the month, while the current value of factor return (which is our dependent variable, X) is calculated at the end of that month. The purpose of conducting this regression analysis is to be able to use it in future. For example, in future, suppose at the beginning of a month we get to know the bond yield. We use the equation to predict factor return for the month and make investment decisions accordingly. This is only possible if Y is known beforehand as is the case here. So, here the contemporaneous variable Y is defined as a variable that is calculated at the beginning of the month while contemporaneous variable X is predicted for that month.
However, if our definition of contemporaneous variable Y is that it is the variable calculated at the same time we calculate contemporaneous variable X then we have an issue. Say, Y in our example is the end of the month bond yield and X is factor return for that month. Both X and Y are calculated at the same time i.e., the end of the month. So, for future predictions, we will not be able to use this equation because we won’t have a value for variable Y beforehand to predict X. We’ll have Y when we already get to know X. In this case, instead of the contemporaneous variable X, we can take the future variable X i.e., the factor return of next month to be able to utilise the regression equation. So, we use end of the month calculated bond yield Y to predict next month’s i.e., future factor return X. This, unlike the usage of contemporaneous X and Y, is relevant.
So, our definition of the contemporaneous variables matters. Also, the timing of the X and Y variables matters. X can only be predicted by Y if it is known beforehand.