Quant

P-Value for Slopes

use the p-value to evaluate the null hypothesis that a slope coefficient is equal to zero.

The p-value is the smallest level of significance for which the null hypothesis can be rejected. We test the significance of coefficients by comparing the p-value to the chosen significance level:

• If the p-value is less than the significance level, the null hypothesis can be rejected.

• If the p-value is greater than the significance level, the null hypothesis cannot be rejected.

Assumptions underlying a multiple regression model include

• A linear relationship exists between the dependent and independent variables.

• The residuals are normally distributed.

• The variance of the error terms is constant for all observations.

• The residual for one observation is not correlated with that of another observation.

• The independent variables are not random, and there is no exact linear relation between any two or more independent variables.

Goal of QQ plot

Check if residuals are normally distributed

Points deviating from the Q–Q line indicate that residuals are not normally distributed. The plot assesses normality, not correlation. Points below the line indicate that observed values are more negative than expected under normality, while points above the line indicate more positive values than expected.

Functional Form Model Misspecifications

Misspecification	Description	Effect
Omission of important independent variable(s)	Based on economic theory, one or more variables that should have been included are omitted.	Biased and inconsistent regression parameters May lead to serial correlation or heteroskedasticity in the residuals
Inappropriate variable form	The relationship between the dependent and independent variables may be non-linear.	May lead to heteroskedasticity in the residuals
Inappropriate variable scaling	Variables may need to be transformed before estimating the regression.	May lead to heteroskedasticity in the residuals or multicollinearity
Data improperly pooled	Sample has periods of dissimilar economic environments (that should not be pooled).	May lead to heteroskedasticity or serial correlation in the residuals

More on Misspecifications

Problem	What goes wrong	Why it matters
Omitted variable (correlated)	Bias + inconsistency (other variables take credit)	Worst case
Omitted variable (uncorrelated)	Intercept bias only	Less severe
Wrong transformation	Nonlinear → forced linear	Poor fit
Bad scaling	Units distort meaning	Misleading comparisons
Pooling data	Different regimes mixed	Wrong conclusions

robust standard errors

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.

Serialcorrelation

Serialcorrelation, 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.

Positiveserial correlation

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

Negativeserial correlation

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

lagged values and positive serial correlation

lagged values
- if lagged values is inaccurate and mess up the whole model and lead to inconsistency

• if not lagged values and uses external variables then can never be inconssitent

postiive serial correlation

• Type I error - false positive

Durbin-Watson (DW) statistic

Residual serial correlation at a single lag can be detected using the Durbin-Watson (DW) statistic.

Breusch-Godfrey (BG) test

more general test (which can accommodate serial correlation at multiple lags) is the Breusch-Godfrey (BG) test.

robuststandard errors

To correct for serial correlation in regression residuals, we can calculate robuststandard 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.

most common sign of multicollinearity

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 R² 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.

What test to check Heteroskedasticity?

Breusch Pagan Chi²

• regress residuals² back into the formula

• BP = n * R² for residuals

  • df = k

• H0: Homo Ha: Hetero

How to fix Hetero?

White-Correlated/Robert Standard error

What test for serial correlation

Durbin Watson - single

Breusch Godfrey - multi

• regress residuals but with lagged variables

  • test if lagged variable slopes is stat sig

• H0: no serial Corr Ha: serial corr

How to fix serial corr?

Newey-West/Robust SE

Multicollinearity

inflated SE

Type II

regress the variables with each other removing 1 at a time

VIF = 1/1-R²

low is good high is bad

high-leverage points

high-leverage points are the extreme observations of the independent (or 'X') variables.

The sum of the individual leverages for all observations is k + 1 (with intercept). If a variable's leverage is higher than three times the average, [3(k + 1) / n], it is considered potentially influential.

studentized residuals

We can identify outliers using the studentized residuals. The following steps outline the procedure:

1. Estimate the regression model using the original sample of size n. Delete one observation and re-estimate the regression using (n - 1) observations. Perform this sequentially, for all observations, deleting one observation at a time.

2. Compare the actual Y value of the deleted observation i to the predicted Y-values using the model parameters estimated with that observation deleted.

   e*i = Yi -∧Y*i

3. The studentized residual is the residual in Step 2 divided by its standard deviation.

   t*i = e*is*e

4. We can then compare this studentized residual to critical values from a t-distribution with n – k – 2 degrees of freedom (because we now only have n – 1 observations), to determine if the observation is influential.

qualitative dependent variable

Financial analysis often calls for the use of a model that has a qualitative dependent variable—a categorical variable, usually a binary variable, which takes on a value of either zero or one. An example of an application requiring the use of a qualitative dependent variable is a model that attempts to estimate the probability of default for a bond issuer. In this case, the dependent variable may take on a value of one in the event of default and zero in the event of no default.