# CS代考程序代写 The Multivariate Linear Regression Analysis and Inference

The Multivariate Linear Regression Analysis and Inference
Zhenhao Gong University of Connecticut

Welcome 2
This course is designed to be:
1. Introductory
2. Leading by interesting questions and applications 3. Less math, useful, and fun!
Most important:
Feel free to ask any questions! 
Enjoy! 

Multivariate regression analysis

Unbiasedness and Consistency 4
There are two terms that are often used to decide whether an estimator is good or not:
􏰀 Unbiasedness: An estimator is unbiased if, the mean of the sampling distribution of the estimator is equal to the true parameter value.
􏰀 Consistency: An estimator is consistent if, as the sample size increases, the sampling distribution of the estimator becomes increasingly concentrated at the true parameter value.

Omitted Variable Bias 5
The OLS estimator will have omitted variable bias when two conditions are true:
􏰀 When the omitted variable is a determinant of the dependent variable
􏰀 When the omitted variable is correlated with the included regressor
Remark: Omitted variable bias means that the first least squares assumption, E(ui|Xi) = 0, is incorrect.

Bias Formula 6
Let the correlation between Xi and ui be corr(Xi, ui) = ρxu. Then the OLS estimator has the limit
βˆ →p β + ρ σ u . 1 1 xuσx
That is, as the sample size increases, βˆ1 is close to β1 + ρxu σu
with increasingly high probability (βˆ1 is biased and inconsistent).
σx

Summary 7
􏰀 Omitted variable bias is a problem whether the sample size is large or small.
􏰀 Whether this bias is large or small in practice depends on the correlation ρxu between the regressor and the error term. The larger |ρxu| is, the larger the bias.
􏰀 The direction of the bias in βˆ1 depends on whether X and u are positively or negatively correlated.
Question: What can we do about omitted variable bias?

The Multiple Regression Model 8
Consider the case of two regressor: Yi=β0+β1X1i+β2X2i+ui, i=1,2,···,n
􏰀 X1, X2 are the two independent variables (regressors)
􏰀 β0 = unknown population intercept
􏰀 β1 = effect on Y of a change in X1, holding X2 constant 􏰀 β2 = effect on Y of a change in X2, holding X1 constant 􏰀 ui = the regression error (omitted factors)

The OLS Estimators 9
The OLS estimators βˆ0, βˆ1, and βˆ2 solves: n
min 􏰃u2i β0,β1,β2 i=1
n
= min 􏰃[Yi − (β0 + β1X1i + β2X2i)]2. β0,β1,β2 i=1

Measures of Fit 10
Three commonly used summary statistics in multiple regression are the standard error of the regression (SER), the regression R2, and the adjusted R2 (as known as R ̄2)