We will be interested in forecasting R_t as a function of lagged information Z_t-1. It is logical to start with a linear regression model. Later we discuss the generalization of this linear model using nonparametric density estimation techniques.
The linear regression model is with a single explanatory variable:
Rt = d0(Z0) + d1(Z1,t-1) + residualt [1]
where d0, d1 are regression coefficients.
This is often presented as
Rt = d0 + d1(Z1,t-1) + residualt [2]
The d0 is interpreted as the intercept and the d1
as the slope coefficient. Equation [1] and [2] are identical. Remember
we have a single explanatory variable. It turns out, in the standard implementation,
of regression, that the Z contains two variables: Z1
might be an interest rate level and Z0 is a constant vector of ones.
In a spread sheet, one can think of the first column as the returns, say from
January 1970 through December 2003, the second column has a "1" in every row,
the third column is the interest rate from December 1969 through to November
2003 (it is lagged). Notice I have no time subscript on Z0 because
it is just a column of ones.
Suppose we ran the following regression:
Rt = d0(Z0) + residualt [3]
This is a regression on the column of ones. What is d0 in this
case? It is just the average return. It is also an equally weighted average
return. According to regression theory, the coefficient is
d0 = INV(Z'Z)Z'R [4]
where Z is just a column of ones. This can be broken down into two
parts.
1
INV(Z'Z) = INV(#obs) = ----
#obs
Z'R = SUM(returns)
Hence, it is obvious that the d0 is the average return, i.e. the
sum divided by the number of observations!
Why are we focussing on this trivial regression? Well, the traditional style of asset management uses average returns (as well as variances and covariances) the mean-variance optimization. Sometimes, moving-window averages (MA) are used, say the last five years. In this case, Z0 would have zeros in the initial rows and "1"s in the last 60 rows (assuming monthly data is used). Sometimes, exponentially weighted moving averages (EWEMA) are used. Again, we can set the Z0 to handle this.
What is the R-square of this regression in [3]. Remember, the definition of R-square is the variance of the regression fitted values divided by the variance of the dependent variable. An R-square of 1.0 or 100% implies that the fitted values exactly coincide with the realized returns.
Var(fitted) Var(d_0)
R-square = ------------ = -------- = 0
Var(R) Var(R)
The R-square is zero. Why? The variance of a constant, d0, is
exactly zero. Remember definition of variance. It is the squared
deviation of the variable from its average. Since the variable is always
equal to one, there is no variance.
Another way of looking at this exercise is to note that those using this style of model are assuming that no other Z variable influences future returns. In fact, in running this special regression (and, indeed, you do not need to run a regression, you simply need to push the average button), they are assuming the d1 and other coefficients are exactly equal to zero.
Using the average as a forecast forces the asset manager implement a strategy with a zero R-square. This is not necessarily a desirable strategy. Indeed, it implies that no other information affects expected returns. It implies that expected returns are constant (at least over the 60-month window of the MA).
Using a more general regression model, we can incorporate predictability. We can execute statistical tests to ensure that the predictability is genuine rather than an artifact of data snooping. The Research Protocol details procedures that avoid potential misspecification.