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СКАЧАТЬ rel="nofollow" href="#equation3_86">Equation 3.86 as a univariate model.

      In Equation 3.86, α and β are constants. In the univariate model, α is typically referred to as the intercept, and β is often referred to as the slope. β is referred to as the slope because it measures the slope of the solid line when Y is plotted against X. We can see this by taking the derivative of Y with respect to X:

      (3.87)

      The final term in Equation 3.86, ϵ, represents a random error, or residual. The error term allows us to specify a relationship between X and Y, even when that relationship is not exact. In effect, the model is incomplete, it is an approximation. Changes in X may drive changes in Y, but there are other variables, which we are not modeling, which also impact Y. These unmodeled variables cause X and Y to deviate from a purely deterministic relationship. That deviation is captured by ϵ, our residual.

      In risk management this division of the world into two parts, a part that can be explained by the model and a part that cannot, is a common dichotomy. We refer to risk that can be explained by our model as systematic risk, and to the part that cannot be explained by the model as idiosyncratic risk. In our regression model, Y is divided into a systematic component, α + βX, and an idiosyncratic component, ϵ.

      (3.88)

      Which component of the overall risk is more important? It depends on what our objective is. As we will see, portfolio managers who wish to hedge certain risks in their portfolios are basically trying to reduce or eliminate systematic risk. Portfolio managers who try to mimic the returns of an index, on the other hand, can be viewed as trying to minimize idiosyncratic risk.

      EVALUATING THE REGRESSION

      Unlike a controlled laboratory experiment, the real world is a very noisy and complicated place. In finance it is rare that a simple univariate regression model is going to completely explain a large data set. In many cases, the data are so noisy that we must ask ourselves if the model is explaining anything at all. Even when a relationship appears to exist, we are likely to want some quantitative measure of just how strong that relationship is.

      Probably the most popular statistic for describing linear regressions is the coefficient of determination, commonly known as R-squared, or just R². R² is often described as the goodness of fit of the linear regression. When R² is one, the regression model completely explains the data. If R² is one, all the residuals are zero, and the residual sum of squares, RSS, is zero. At the other end of the spectrum, if R² is zero, the model does not explain any variation in the observed data. In other words, Y does not vary with X, and β is zero.

      To calculate the coefficient of determination, we need to define two additional terms: TSS, the total sum of squares, and ESS, the explained sum of squares. They are defined as:

      (3.89)

      These two sums are related to the previously encountered residual sum of squares, as follows:

      (3.90)

      In other words, the total variation in our regressand, TSS, can be broken down into two components, the part the model can explain, ESS, and the part the model cannot, RSS. These sums can be used to compute R²:

      (3.91)

      As promised, when there are no residual errors, when RSS is zero, R² is one. Also, when ESS is zero, or when the variation in the errors is equal to TSS, R² is zero. It turns out that for the univariate linear regression model, R² is also equal to the correlation between X and Y squared. If X and Y are perfectly correlated, ρ_xy = 1, or perfectly negatively correlated, ρ_xy = –1, then R² will equal one.

      Estimates of the regression parameters are just like the parameter estimates we examined earlier, and subject to hypothesis testing. In regression analysis, the most common null hypothesis is that the slope parameter, β, is zero. If β is zero, then the regression model does not explain any variation in the regressand.

      In finance, we often want to know if α is significantly different from zero, but for different reasons. In modern finance, alpha has become synonymous with the ability of a portfolio manager to generate excess returns. This is because, in a regression equation modeling the returns of a portfolio manager, after we remove all the randomness, ϵ, and the influence of the explanatory variable, X, if α is still positive, then it is suggested that the portfolio manager is producing positive excess returns, something that should be very difficult in efficient markets. Of course, it's not just enough that the α is positive; we require that the α be positive and statistically significant.

      Sample Problem

      Question:

      As a risk manager and expert on statistics, you are asked to evaluate the performance of a long/short equity portfolio manager. You are given 10 years of monthly return data. You regress the log returns of the portfolio manager against the log returns of a market index.

      Assume both series are normally distributed and homoscedastic. From this analysis, you obtain the following regression results:

      What can we say about the performance of the portfolio manager?

      Answer:

      The R² for the regression is low. Only 8.11 percent of the variation in the portfolio manager's returns can be explained by the constant, beta, and variation in the market. The rest is idiosyncratic risk, and is unexplained by the model.

      That said, both the constant and the beta seem to be statistically significant (i.e., they are statistically different from zero). We can get the t-statistic by dividing the value of the coefficient by its standard deviation. For the constant, we have:

      Similarly, for beta we have a t-statistic of 2.10. Using a statistical package, we calculate the corresponding probability associated with each t-statistic. This should be a two-tailed test with 118 degrees of freedom (10 years × 12 months per year – 2 parameters). We can reject the hypothesis that the constant and slope are zero at the 2 percent level and 4 percent level, respectively. In other words, there seems to be a significant market component to the fund manager's return, but the manager is also generating statistically significant excess returns.

      Linear Regression (Multivariate)

      Univariate regression models are extremely common in finance and risk management, but sometimes we require a slightly more complicated model. In these cases, we might use a multivariate regression model. The basic idea is the same, but instead of one regressand and one regressor, we have one regressand and multiple regressors. Our basic equation will look something like:

      (3.92)СКАЧАТЬ