Regression Hal Varian 10 April 2006

What is regression? History Curve fitting v statistics Correlation and causation Statistical models Gauss-Markov theorem Maximum likelihood Conditional mean What can go wrong… Examples

Francis Galton, 1877 Plotted first regression line: Diameter of sweetpeas v diameter of parents Heights of fathers v heights of sons Sons of unusually tall fathers tend to be tall, but shorter than their fathers. Galton called this “regression to mediocrity”. But this is also true the other way around! Regression to the mean fallacy. Pick the lowest scoring 10% on the midterm and give them extra tutoring If they do better on the final, what can you conclude? Did the tutoring help?

Regression analysis Assume a linear relation between two variables and estimate unknown parameters y t = a + b x t + e t for t= 1,…,T observed = fitted + error or residual dependent variable ~ independent variables/predictors/correlates

Curve fitting v regression Often choose (a,b) to minimize the sum of squared residuals (“least squares”) Why not absolute value of residuals? Why not fit x t = a + b y t ? How much can you trust the estimated values? Need a statistical model to answer these questions! Linear regression: linear in parameters Nonlinear regression, local regression, general linear model, general additive model: same principles apply

Possible goals Estimate parameters (a, b and error variance) Test hypotheses (such as “x has no influence on y”) Make predictions about y conditional on observing a new x-value Summarize data (most common unstated goal!)

Summarizing relationships Would like to be able to interpret regression as “causal” “If x changes by  x, then y will on average change by  x b.” Correlation v causation Compare the time on my wristwatch with the time on your wristwatch… Even ideally, best you can say is: “When x changes by  x in the sample, then on average y changes by  x b in the sample.”

Problem with causality There may be a “third cause” “my watch time” and “your watch time” both depend on NIST time Economics example income ~ b education + (unobserved IQ+other) education ~ IQ Higher income is associated with higher education in sample, but b is a biased estimate of partial effect of education on income Need a controlled experiment or more elaborate estimation technique to resolve this “simultaneous equations bias”

Statistical regression model y t = a + b x t +  t for t= 1,…,T Think of random variable  t as the sum of the other omitted effects What are attractive properties for error term? E  t = 0 Var  t = constant E  t  s  0 (errors are independent) E x t  t = 0 (errors are conditionally uncorrelated with explanatory variables – often problematic for reasons on last slide! Exogenous v endogenous.) Have to ask: how do the variables you don’t observe affect the variables you do observe?

Optimality properties Gauss-Markov theorem: If the error term has these properties, then the linear regression estimates of (a,b) are BLUE = “best linear unbiased estimates” = out of all unbiased estimates that are linear in y t the least squares estimates have minimum variance. If  t are Normal IID distributed, then the OLSQ estimates are maximum likelihood estimates

Conditional means In the regression model, note that the expected value of y t is a + b x t. So the conditional mean is linear in x t, which is another interpretation of regression. More generally, can think of regression model as being: E y t = f(x t, b)

Regression output Estimates of parameters Standard errors of estimates and error term t-statistics = estimate/se and p-values R 2 = goodness of fit measure Total SS = Fitted SS + Residual SS R 2 = Fitted SS / Total SS

Example from R > x <- 1:100 > y <- x + 10*rnorm(100) > summary(lm(y~x)) Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) x <2e-16 *** Residual standard error: on 98 degrees of freedom Multiple R-Squared: ,Adjusted R- squared :

What can go wrong? Nonlinear relationship Try quadratic, interaction term, logs, etc. Var  t is not constant Heteroskedasticity – affects testing not estimates Take logs or use weighted least squares Serial correlation – affects testing and prediction accuracy Use time series methods Multiple regression – colinearity Socks ~ right shoes + left shoes + shoes + error

What can go wrong, cont Errors in variables Underestimate magnitude of true effect Omitted variable bias Bias depending on correlation of omitted with included variables Simultaneous equations bias Third cause alluded to earlier, need to estimate full model or use controlled experiment Outliers Non-normality of errors and influential observations – remove them or use robust estimation

Diagnostics Look at residuals!! R allows you to plot various regression diagnostics reg <- lm(y~x) plot(reg) Examples to follow…