Week 6: Multiple Regression and Inference (02/16/2023)

Ozlem Tuncel

otuncelgurlek1@gsu.edu

⚠️ CAUTION: DO NOT SOLELY RELY ON MY NOTES. THERE MIGHT BE TYPOS AND MISTAKES. ALWAYS TAKE YOUR NOTES!

✔️ The goal of this week is to learn multiple OLS and inference.

Here are some key points:

• Just like bivariate regression, now we have multiple independent variables.
• $\beta = ({X^\prime} X)^{-1} {X^\prime}Y$ - knowing this by heart will help you with proofs
• Gauss-Markov Assumptions
  1. Linearity - our parameters (our betas) are linear!
  2. No Perfect Multicollinearity (Full Rank) (perfect multicollinearity is when your variables are strongly correlated) and N>K (number of observations should be bigger than number of variables.

  3. E(u)=0 Conditional Mean Zero assumption - so our average error term should be zero. Xs are exogenous => E(u

     X) = 0. This assumption fails if X and u are correlated.

  4. Spherical Disturbances (Homoskedasticity and Autocorrelation) autocorrelation mostly deals with time and homoskedasticity mostly deals with space. Our error terms are not correlated with each other under homoskedasticity (within-group correlation will cause heteroskedasticity). And, there is no correlation between the error term and independent variables.
  5. Non-stochastic Regressors - we are measuring things that vary but we assume that they are fixed. $Cov(X, u) = 0$. This assumption also assumes that we do not have measurement error.
  6. Normality or $u \sim N[0, \sigma^2 I]$
• Violations of most assumptions, won’t bias our coefficient estimates, but we will have no clue about the precision of our estimates because our standard errors will be way off.
• Consistencty is asymptotic (aka large sample size) characteristics.
• Goodness-of-fit - overall quality of our model. Here we are going to talk about $R^2$, $R^2_{adj}$, and $F-statistic$. How well my model fit the data?
  1. F-statistic: depends on the degrees of freedom and sample size. Bivariate regression has 2 degrees of freedom. Comparing the model (unrestricted model) to some other model (restiricted model). If you have not significant F-stat - that is bad!
  2. $R^2$: people give too much importance to this - intuitively, this is the measure of amount of variance explained by your variables. Increases as we add variables to our regression.
  3. $R^2_{adj}$: much better than R-squared. It penalizes for the number of k.
• You’ll always gave omitted variable bias - this is due to human nature. It is impossible account for all factors that explain an outcome. Overfitting is a much worse problem.
• Interpretation: independent effects, so we can interpret each coefficient. We can say “one unit increase in our X increases Y by XXXX”
• Standardized coefficients: helps us to identify the effect when we have two different scales. Now, we say “one standard deviation increase in our X increases Y by XXXX standard deviations”. But, this makes substantive interpretation harder, right?