Model Specification

Quantitative Methods 12 min readModule 3 of 4Save

What you'll learn

LOS 1.g
Describe how model misspecification affects the results of a regression analysis and how to avoid common forms of misspecification.
LOS 1.h
Explain the types of heteroskedasticity and how it affects statistical inference.
LOS 1.i
Explain serial correlation and how it affects statistical inference.
LOS 1.j
Explain multicollinearity and how it affects regression analysis.

Warm Up: Model Specification

Regression model specification is the selection of the explanatory (independent) variables to be included in a model, and the transformations (if any) of those explanatory variables.

For example: suppose that we’re trying to predict a P/E ratio using a cross sectional regression based on fundamental variables that are related to P/E. Valuation theory suggests that variables associated with P/E include: the stock’s dividend payout ratio (DPO), growth rate (G), and beta (B).

The principles of model specification require that the chosen independent variables have an economic rationale, that the model is parsimonious (i.e., simple and efficient), and that the regression’s functional form is correctly specified. A properly specified model should also perform well outside the sample, and not violate key regression assumptions. Figure 1.3 summarizes different types of functional form misspecifications.

Figure 1.3: Functional Form Model Misspecifications

Examples of Misspecification of Functional Form

Let’s start with a regression in which we’re trying to predict monthly returns on a portfolio of Chinese stocks (R) using four independent variables: portfolio beta (B), the natural log of market capitalization (lnM), the natural log of the price to book ratio ln(PB), and free float (FF). Free float is equal to the ratio of shares available for trading by the investing public to total number of outstanding shares. The regression is estimated with 72 monthly observations from July 1996 to June 2002.

The correct specification of the model is as follows:

R = b_{0} + b_{1} β + b_{2} ln M + b_{3} ln P B + b_{4} F F + ε

Misspecification #1: Omitting a Variable Suppose we do not include lnM in the regression model:

R = a_{0} + a_{1} β + a_{2} ln P B + a_{3} F F + ε

If lnM is correlated with any of the remaining independent variables (β, lnPB, or FF), then the error term is also correlated with the same independent variables, and the resulting regression coefficients (the estimates of $a_{0}$ , $a_{1}$ , $a_{2}$ , and $a_{3}$ ) will be biased and inconsistent. That means our hypothesis tests and predictions using the model will be unreliable.

If the omitted variable is uncorrelated with the other independent variables, then the error term will reflect the value of the missing variable’s contribution, and the value of the intercept ( $a_{0}$ ) will be biased (even though the estimates of $a_{1}$ , $a_{2}$ , and $a_{3}$ will be correct).

Misspecification #2: Variable Should Be Transformed

Regression assumes that the dependent variable is linearly related to each of the independent variables. Initially, however, the relationship between the model inputs and output may be non linear. For example, the relationship between market capitalization and portfolio returns is not linear; rather the natural log of market cap is linearly related to returns.

If we include market cap in our regression without transforming it by taking the natural log i.e., if we use M and not ln(M), we’ve misspecified the model.

R = c_{0} + c_{1} β + c_{2} M + c_{3} ln P B + c_{4} F F + ε

Misspecification #3: Inappropriate Scaling of the Variable If financial statement data are included in the regression model, a common transformation is to standardize the variables by dividing by sales (for income statement or cash flow items) or total assets (for balance sheet items). You should recognize these as items from common size financial statements. Examples of other scaling transformations include squaring the variable or taking the square root of the variable.

Suppose that rather than defining FF as the proportion of shares available to be traded, we instead use the number of shares available to be traded. In such a case, we could have similar companies with very different values for FF; an example of inappropriate scaling.

Misspecification #4: Incorrectly Pooling Data

Suppose the relationship between returns and the independent variables during the first three years is actually different than the relationship in the second 3 year period (i.e., the regression coefficients are different from one time period to the next). By pooling the data and estimating one regression over the entire 6 year period, rather than estimating separate regressions over each of the two subperiods, we have misspecified the model. Our hypothesis tests and predictions of portfolio returns will be misleading.

Warm Up: Violation of Regression Assumptions

Regression analysis relies on the assumptions listed earlier. When these assumptions are violated, the inferences drawn from the model are questionable. There are three primary assumption violations that you will encounter: (1) heteroskedasticity, (2) serial correlation (i.e., autocorrelation), and (3) multicollinearity.

On exam day, you want to be able to answer the following four questions about each of the three assumption violations:

What is it? What is its effect on regression analysis? How do we detect it?
How do we correct for it?

What Is Heteroskedasticity?

Recall that one of the assumptions of multiple regression is that the variance of the residuals is constant across observations. Heteroskedasticity occurs when the variance of the residuals is not the same across all observations in the sample. This happens when there are subsamples that are more spread out than the rest of the sample.

Unconditional heteroskedasticity occurs when the heteroskedasticity is not related to the level of the independent variables, which means that it doesn’t systematically increase or decrease with changes in the value of the independent variable(s). While this is a violation of the equal variance assumption, it usually causes no major problems with the regression.

Conditional heteroskedasticity is heteroskedasticity that is related to (i.e., conditional on) the level of the independent variables. For example, conditional heteroskedasticity exists if the variance of the residual term increases as the value of the independent variable increases, as shown in Figure 1.4. (Notice that in this figure the residual variance associated with the larger values of the independent variable, X, is larger than the residual variance associated with the smaller values of X.) Conditional heteroskedasticity does create significant problems for statistical inference.

Figure 1.4: Conditional Heteroskedasticity

Effect of Conditional Heteroskedasticity on Regression Analysis There are two effects of conditional heteroskedasticity that you should be aware of:

The standard errors are usually unreliable estimates. (For financial data, these standard errors are usually underestimated, resulting in Type I errors.) The F test for the overall model is also unreliable.
Coefficient estimates are still consistent and unbiased.

Detecting Conditional Heteroskedasticity

There are two methods to detect conditional heteroskedasticity: by examining scatter plots of the residuals and by using the Breusch Pagan chi square ( $χ^{2}$ ) test. A scatter plot of the residuals versus one or more of the independent variables can reveal patterns among observations.

Worked example: Detecting conditional heteroskedasticity with a residual plot

You have been studying five years of monthly returns for a particular mutual fund, hoping to draw conclusions about the fund’s average performance. You calculate the mean return, the standard deviation, and the fund’s beta by regressing the fund’s returns on S&P 500 index returns (the independent variable). The standard deviation of returns and the fund’s beta don’t seem to fit the firm’s stated risk profile.

For your analysis, you have prepared a scatter plot of the error terms (actual return minus predicted return) for the regression using the five years of returns, as shown next. Determine whether the residual plot indicates that there may be a problem with the data.

Residual Plot

Solution

The residual plot indicates the presence of conditional heteroskedasticity. Notice how the variation in the regression residuals increases as the independent variable increases: this indicates that the variance of the fund’s returns about the mean is related to the level of the independent variable.

More formally, we can use the Breusch Pagan (BP) test to detect conditional heteroskedasticity. The BP test calls for the squared residuals (as the dependent variable) to be regressed on the original set of independent variables. If conditional heteroskedasticity is present, the independent variables will significantly contribute to the explanation of the variability in the squared residuals.

The test statistic for the Breusch Pagan test, which has a chi square ( $χ^{2}$ ) distribution, is calculated as:

BP = n R_{resid}^{2} \sim χ_{k}^{2}

This is a one tailed test, because heteroskedasticity is only a problem if the $R^{2}$ and the BP test statistic are too large.

Worked example: The Breusch Pagan test

Suppose that the residual plot of mutual fund returns over time shows evidence of heteroskedasticity. To confirm our suspicions, we regress the squared residuals from the original regression on the independent variable, the S&P 500 index returns. The $R^{2}$ from that regression is 8%.

How would we use the Breusch Pagan test to determine whether heteroskedasticity is present at the 5% significance level?

Solution

With five years of monthly observations, n is equal to

The test statistic is:

n R^{2} = 60 (0.08) = 4.8

The one tailed critical value for a chi square distribution with one degree of freedom and α of 5% is 3.841. Therefore, we should reject the null hypothesis and conclude that we have a problem with conditional heteroskedasticity.

Correcting Heteroskedasticity

To correct for conditional heteroskedasticity of regression residuals, we can calculate robust standard errors (also called White corrected standard errors or heteroskedasticity consistent standard errors). These robust standard errors are then used to recalculate the t statistics using the original regression coefficients for hypothesis testing.

Serial correlation, also known as autocorrelation, refers to a situation in which regression residual terms are correlated with one another; that is, not independent. Serial correlation can pose serious problem with regressions using time series data.

Positive serial correlation exists when a positive residual in one time period increases the probability of observing a positive residual in the next time period.

Negative serial correlation occurs when a positive residual in one period increases the probability of observing a negative residual in the next period.

Effect of Serial Correlation on Model Parameters Consider a model that employs a lagged value of the dependent variable as one of the independent variables. Residual autocorrelation in such a model causes the estimates of the slope coefficients to be inconsistent. If the model does not have any lagged dependent variables, then the estimates of the slope coefficient will be consistent.

Effect on Standard Errors Positive first order serial correlation in residuals is most common in financial time series data. Positive serial correlation typically results in coefficient standard errors that are too small, causing the computed t statistics (and F statistic) to be larger than they should be, which will lead to Type I errors.

Detecting Serial Correlation

Residual serial correlation at a single lag can be detected using the Durbin Watson (DW) statistic. A more general test (which can accommodate serial correlation at multiple lags) is the Breusch Godfrey (BG) test. The BG test regresses the regression residuals against the original set of independent variables, plus one or more additional variables representing lagged residual(s):

ε^_{t} = b_{0} + a_{1} X_{1 t} + a_{2} X_{2 t} + \dots + a_{k} X_{k t} + p_{1} ε_{t - 1} + p_{2} ε_{t - 2}

We then test whether each of the slope coefficients of the lagged residuals is statistically significantly different from 0.

H_{0} : p_{1} = 0 vs. H_{a} : p_{1} \neq = 0

The BG test statistic has an F distribution with p and n − p − k − 1 degrees of freedom, where p = the number of lags tested. Most software packages provide the BG test statistic. If the BG test statistic > critical value, we reject the null hypothesis of no serial correlation.

Correction for Serial Correlation

To correct for serial correlation in regression residuals, we can calculate robust standard errors (also called Newey West corrected standard errors or heteroskedasticity consistent standard errors). These robust standard errors are then used to recalculate the t statistics using the original regression coefficients.

Multicollinearity refers to the condition when two or more of the independent variables, (or linear combinations of three or more independent variables), in a multiple regression are highly correlated with each other. This condition inflates standard errors and lowers t stats.

Effect of Multicollinearity on Model Parameters Multicollinearity does not affect the consistency of slope coefficients, but makes those estimates imprecise and unreliable.

Effect on Standard Errors

Standard errors are too high because we cannot assign precise credit for explaining the variation in the dependent variable to the independent variables. This leads to Type II errors.

Detection The most common sign of multicollinearity is when t tests indicate that none of the individual coefficients are significantly different than zero, but the F test indicates that at least one of the coefficients is statistically significant, and the $R^{2}$ is high. This suggests that the variables together explain much of the variation in the dependent variable, but the individual independent variables don’t. This can happen when the independent variables are highly correlated with each other so while their common source of variation is explaining the dependent variable, the high degree of correlation also “washes out” the individual effects.

More formally, we can quantify multicollinearity using the variance inflation factor (VIF) for each of the independent variables. We start by regressing one of the independent variable “j” against the remaining independent variables. The $R_{j}^{2}$ from that equation is then used to calculate the VIF of that variable.

VIF_{j} = \frac{1}{1 - R _{j}^{2}}

High values of $R_{j}^{2}$ signal that the variable is well explained by other variables, and indicates that the variable will have a high VIF. A $V I F_{j}$ value of 1 (i.e., $R^{2}$ = 0) indicates that the variable j is not highly correlated with other independent variables. VIF values greater than 5 (i.e., $R^{2}$ > 80%) warrant further investigation, while values above 10 (i.e., $R^{2}$ > 90%) indicates severe multicollinearity.

Correction The most common method to correct for multicollinearity is to omit one or more of the correlated independent variables. Alternatively, we can use a different proxy for one of the included independent variables, or increase the sample size.

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