Week 7: Multiple Regression PDF i.e. analogous to the

Calculation of Multiple Regression with Three. Independent Variables Using a Programable Pocket. Calculator. Paul Evenson. Follow this and additional works at

The Simple Linear Regression Model

A Linear Probabilistic Model. Page 9. 9. The only randomness on the right-hand side of the linear model equation is in ε. (x is fixed!) What is μY

Multiple Regression

We'd never try to find a regression by hand and even Assuming that the conditions for multiple regression are met

Simple Linear Regression

The fitted (or predicted) values are obtained by substituting x1. …

Testing Mediation with Regression Analysis

coefficient from a multiple regression. The indirect effect is the in R run the separate regression models described above calculate the indirect effect ...

Predicting Hand Surface Area from a Two-Dimensional Hand Tracing

Nov 3 2017 By applying a linear regression formula between the 2D and 3D areas

Latent Topic Analysis of the Post Property for Sales to Predict a

prediction model of second-hand condominium using multiple linear regression and artificial • Calculate the accuracy of the multiple regression model based ...

Long Hand Regression in R

Linear Regression via Least Squares. So in the following we will be Instead you compute the equivalent of 1 over the matrix

The Omission or Addition of an Independent Variate in Multiple

variates the equations to determine the linear regression coefficients bl numbers in the right-hand column is then calculated. The sum of. Page 5. 1938 ...

The Statistical Software Toolkit: BERDC Seminar Series 1

Multiple linear regression. Yes but hard linear-regression-by-hand/. [17] https://owlcation.com/stem/How-to-Create-a-Simple-Linear-Regression-Equation.

Calculation of Multiple Regression with Three Independent

Evenson Paul

Multiple Regression

We'd never try to find a regression by hand and Second

Interactions in Multiple Linear Regression

On the other hand if men

Week 7: Multiple Regression

i.e. analogous to the simple linear regression case! Disclaimer: the final equation is exactly true for all non-intercept coefficients if you remove the.

The Simple Linear Regression Model

The only randomness on the right-hand side of the linear model equation is in ?. (x is fixed!) What is ?Y

Testing Mediation with Regression Analysis

Psy 523/623 Structural Equation Modeling Spring 2020 Step 4 Conduct a multiple regression analysis with X and M predicting.

Multiple Regression - Introduction

The formula for the SEE is the same as in the bivariate case; however K = 2 in this example

Linear Regression using Stata

Technically linear regression estimates how much Y changes when X changes one unit. In Stata use the command regress

Multiple Regression - Prediction & Confidence Intervals Demystified

With multivariate regression the confidence and prediction interval must account for the simultaneous wiggle of multiple X variables. – To calculate the

Review of Multiple Regression

Jan 3 2022 Before doing other calculations

Multiple Linear Regression by Hand (Step-by-Step) - Statology

18 nov 2020 · This tutorial explains how to perform multiple linear regression by hand including a step-by-step example

[PDF] Calculation of Multiple Regression with Three Independent

This paper describes a multiple re- gression program for an equation with one dependent and three independent variables which was written for a Hewlett-

[PDF] Multiple Regression - UC Berkeley Statistics

We'd never try to find a regression by hand and even calculators aren't really up to the task This is a job for a statistics program on a computer

[PDF] Multiple Linear Regression

All standard statistical software packages compute and show the standard deviations of the regression coefficients Inference concerning a single ?i is based on

[PDF] Multiple Regression 1

The answer depends upon the use to which we wish to put the estimated regression equation The issue is whether the equation is to be used simply for predicting

[PDF] Multiple Regression - Introduction

The formula for the SEE is the same as in the bivariate case; however K = 2 in this example since there are two independent variables MSE = 1 - K - N SSE

[PDF] Multiple Linear Regression (2nd Edition) Mark Tranmer Jen Murphy

The simple linear regression line plot in Figure 5 shows an R2 value of 0 359 at the top right hand side of the plot This means that the variable social

[PDF] Chapter 3 Multiple Linear Regression Model - IIT Kanpur

independent variables called a multiple linear regression model This model generalizes the simple linear regression in two ways

[PDF] Multiple Linear Regression - San Jose State University

Multiple Linear Regression Remark This is the same formula for ˆ ? = (ˆ?0 ˆ ?1) in simple linear regression To demonstrate it consider the toy data

18 nov. 2020 · This tutorial explains how to perform multiple linear regression by hand, including a step-by-step example.

How do you manually calculate multiple regression?
With these variables, the usual multiple regression equation, Y = a + b₁X₁ + b₂X₂, becomes the quadratic polynomial Y = a + b₁X + b₂X². This is still considered a linear relationship because the individual terms are added together.
What is the formula for calculating multiple regression?
Therefore, the formula for calculation is Y = a + bX + E, where Y is the dependent variable, X is the independent variable, a is the intercept, b is the slope, and E is the residual. Regression is a statistical tool to predict the dependent variable with the help of one or more independent variables.
How do you manually calculate regression?
The five steps to follow in a multiple regression analysis are model building, model adequacy, model assumptions – residual tests and diagnostic plots, potential modeling problems and solution, and model validation.

Week 7: Multiple Regression

Brandon Stewart

Princeton

October 22, 24, 20181

These slides are heavily in

uenced by Matt Blackwell, Adam Glynn, Justin Grimmer,

Jens Hainmueller and Erin Hartman.

Stewart (Princeton)Week 7: Multiple RegressionOctober 22, 24, 2018 1 / 140

Where We've Been and Where We're Going...

Last Week

I regression with two variables Iomitted variables, multicollinearity, interactionsThis Week

IMonday:

F matrix form of linear regression Ft-tests, F-tests and general linear hypothesis testsI

Wednesday:

F problems withp-values

Fagnostic regression

Fthe bootstrapNext Week

I break!

Ithen:::diagnosticsLong Run

I probability!inference!regression!causal inference

Questions?

Stewart (Princeton)Week 7: Multiple RegressionOctober 22, 24, 2018 2 / 140

1Matrix Form of Regression

2OLS inference in matrix form

3Standard Hypothesis Tests

4Testing Joint Signicance

5Testing Linear Hypotheses: The General Case

6Fun With(out) Weights

7Appendix: Derivations and Consistency

8The Problems withp-values9Agnostic Regression

10Inference via the Bootstrap

11Fun With Weights

12Appendix: Trickyp-value ExampleStewart (Princeton)Week 7: Multiple RegressionOctober 22, 24, 2018 3 / 140

The Linear Model with New Notation

Remember that we wrote the linear model as the following for all i2[1;:::;n]:y i=0+xi1+zi2+uiImagine we had annof 4. We could write out each formula:y

1=0+x11+z12+u1(unit 1)y

2=0+x21+z22+u2(unit 2)y

3=0+x31+z32+u3(unit 3)y

4=0+x41+z42+u4(unit 4)Stewart (Princeton)Week 7: Multiple RegressionOctober 22, 24, 2018 4 / 140

The Linear Model with New Notation

1=0+x11+z12+u1(unit 1)

2=0+x21+z22+u2(unit 2)

3=0+x31+z32+u3(unit 3)

4=0+x41+z42+u4(unit 4)We can write this as:

2 6 64y
1 y 2 y 3 y 43
7 75=2
6 641
1 1 13 7 750+2
6 64x
1 x 2 x 3 x 43
7 751+2
6 64z
1 z 2 z 3 z 43
7 752+2
6 64u
1 u 2 u 3 u 43
7

75Outcome is alinea rcombination of the the x,z, anduvectorsStewart (Princeton)Week 7: Multiple RegressionOctober 22, 24, 2018 5 / 140

Grouping Things into Matrices

Can we write this in a more compact form?

Yes!

Let Xandbe the following:X

(43)=2 6

641x1z1

1x2z2 1x3z3

1x4z43

7 75
(31)=2 4 0 1 23
5 Stewart (Princeton)Week 7: Multiple RegressionOctober 22, 24, 2018 6 / 140

Back to Regression

Xis then(k+ 1) design matrix of independent variablesbe the (k+ 1)1 column vector of coecients.Xwill ben1:X=0+1x1+2x2++kxkWe can compactly write the linear model as the following:

y (n1)=X (n1)+u(n1)We can also write this at the individual level, wherex0iis theith row ofX: y i=x0i+uiStewart (Princeton)Week 7: Multiple RegressionOctober 22, 24, 2018 7 / 140

Multiple Linear Regression in Matrix Form

Let bbe the matrix of estimated regression coecients andbybe the vector of tted values:b =2 6 664b

0b1...

bk3 7 775b
y=XbIt might be helpful to see this again more written out: b y=2 6 664b
y1 b y2... b yn3 7

775=Xb=2

6 6641
b0+x11b1+x12b2++x1Kbk

1b0+x21b1+x22b2++x2Kbk...

1 b0+xn1b1+xn2b2++xnKbk3 7

775Stewart (Princeton)Week 7: Multiple RegressionOctober 22, 24, 2018 8 / 140

Residuals

We can easily write the

residuals in matrix fo rm: b u=yXbOur goal as usual is to minimize the sum of the squared residuals, which we saw earlier we can write:b u0bu= (yXb)0(yXb)Stewart (Princeton)Week 7: Multiple RegressionOctober 22, 24, 2018 9 / 140

OLS Estimator in Matrix Form

Goal: minimize the

sum of the squa redresiduals Take (matrix) derivatives, set equal to 0 (see Appendix)

Resulting rst order conditions:

0(yXb) = 0Rearranging:

0Xb=X0yIn order to isolate

b, we need to move theX0Xterm to the other side of the equals sign.We've learned about matrix multiplication, but what about matrix \division"? Stewart (Princeton)Week 7: Multiple RegressionOctober 22, 24, 2018 10 / 140

Back to OLS

Let's assume, for now, that the inverse ofX0XexistsThen we can write the OLS estimator as the following:

b = (X0X)1X0y\ex prime ex inverse ex prime y"sear it into your soul. Stewart (Princeton)Week 7: Multiple RegressionOctober 22, 24, 2018 11 / 140

Intuition for the OLS in Matrix Form

b = (X0X)1X0yWhat's the intuition here? \Numerator"X0y: is approximately composed of the covariances between the columns ofXandy\Denominator"X0Xis approximately composed of the sample variances and covariances of variables withinXThus, we have something like: b (variance ofX)1(covariance ofX&y)

i.e. analogous to the simple linear regression case!Disclaimer: the nal equation is exactly true for all non-intercept coecients if you remove the

intercept fromXsuch that^0= Var(X0)1Cov(X0;y). The numerator and denominator are the variances and covariances ifXandyare demeaned and normalized by the sample size minus 1. Stewart (Princeton)Week 7: Multiple RegressionOctober 22, 24, 2018 12 / 140

1Matrix Form of Regression

2OLS inference in matrix form

3Standard Hypothesis Tests

4Testing Joint Signicance

5Testing Linear Hypotheses: The General Case

6Fun With(out) Weights

7Appendix: Derivations and Consistency

8The Problems withp-values9Agnostic Regression

10Inference via the Bootstrap

11Fun With Weights

12Appendix: Trickyp-value ExampleStewart (Princeton)Week 7: Multiple RegressionOctober 22, 24, 2018 13 / 140

OLS Assumptions in Matrix Form

1Linearity:y=X+u2Random/iid sample: (yi;x0i) are a iid sample from the population.3No perfect collinearity:Xis ann(k+ 1) matrix with rankk+ 14Zero conditional mean:E[ujX] =05Homoskedasticity: var(ujX) =2uIn6Normality:ujXN(0;2uIn)Stewart (Princeton)Week 7: Multiple RegressionOctober 22, 24, 2018 14 / 140

Assumption 3: No Perfect Collinearity

Denition (Rank)

The rank of a matrix is the maximum nu mberof linea rlyindep endent

columns.In matrix form:Xis ann(k+ 1) matrix with rankk+ 1IfXhas rankk+ 1, then all of its columns are linearly independent...and none of its columns are linearly dependent implies no perfect

collinearityXhas rankk+ 1 and thus (X0X) is invertibleJust like variation inXled us to be able to divide by the variance in

simple OLS Stewart (Princeton)Week 7: Multiple RegressionOctober 22, 24, 2018 15 / 140

Assumption 5: Homoskedasticity

The stated homoskedasticity assumption is: var(ujX) =2uInTo really understand this we need to know what var(ujX) is in full

generality.The variance of a vector is actually a matrix: var[u] = u=2 6

664var(u1) cov(u1;u2):::cov(u1;un)

cov(u2;u1) var(u2):::cov(u2;un) cov(un;u1) cov(un;u2):::var(un)3 7

775This matrix is alwayssymmetric since cov( ui;uj) = cov(uj;ui) by

denition. Stewart (Princeton)Week 7: Multiple RegressionOctober 22, 24, 2018 16 / 140

Assumption 5: The Meaning of Homoskedasticity

What does var(ujX) =2uInmean?I

nis thennidentity matrix,2uis a scalar.Visually: var[u] =2uIn=2 6 664

2u0 0:::0

02u0:::0

0 0 0::: 2u3

775In less matrix notation:

I var(ui) =2ufor alli(constant variance)I

cov(ui;uj) = 0 for alli6=j(implied by iid)Stewart (Princeton)Week 7: Multiple RegressionOctober 22, 24, 2018 17 / 140

Unbiasedness of

Is ^still unbiased under assumptions 1-4? DoesE[^] =?^

X0X1X0y(linearity and no collinearity)^

X0X1X0(X+u)^

X0X1X0X+X0X1X0u^

=I+X0X1X0u^ =+X0X1X0uE[^jX]=E[jX] +E[X0X1X0ujX]E[^jX]=+X0X1X0E[ujX]E[^jX]=(zero conditional mean)So, yes! Stewart (Princeton)Week 7: Multiple RegressionOctober 22, 24, 2018 18 / 140

A Much Shorter Proof of Unbiasedness of

A shorter (but less helpful later) proof of unbiasedness,E[^] =E[X0X1X0y] (denition of the estimator)=

X0X1X0X(expectation of y)=Now we know the sampling distribution is centered onwe want to derive

the variance of the sampling distribution conditional onX.Stewart (Princeton)Week 7: Multiple RegressionOctober 22, 24, 2018 19 / 140

Rule: Variance of Linear Function of Random Vector Recall that for a linear transformation of a random variableXwe have

V[aX+b] =a2V[X] with constantsaandb.We will need an analogous rule for linear functions of random vectors.

Denition (Variance of Linear Transformation of Random Vector) Letf(u) =Au+Bbe a linear transformation of a random vectoruwith non-random vectors or matricesAandB. Then the variance of the transformation is given by:

V[f(u)] =V[Au+B] =AV[u]A0=AuA0Stewart (Princeton)Week 7: Multiple RegressionOctober 22, 24, 2018 20 / 140

Conditional Variance of

^=+ (X0X)1X0uandE[^jX] =+E[(X0X)1X0ujX] =so the OLS

estimator is a linear function of the errors. Thus:V[^jX]=V[jX] +V[(X0X)1X0ujX]=V[(X0X)1X0ujX]= (X0X)1X0V[ujX]((X0X)1X0)0(Xis nonrandom givenX)= (X0X)1X0V[ujX]X(X0X)1= (X0X)1X02IX(X0X)1(by homoskedasticity)=2I(X0X)1X0X(X0X)1=2(X0X)1This gives the (k+ 1)(k+ 1)va riance-covariancematrix of ^.To estimateV[^jX], we replace2with its unbiased estimator ^2, which is now

written using matrix notation as: ^2=P i^u2in(k+ 1)=^u0^un(k+ 1)Stewart (Princeton)Week 7: Multiple RegressionOctober 22, 24, 2018 21 / 140

Sampling Variance for

Under assumptions 1-5, the

va riance-covariancematrix of the OLS estimators is given by:

V[^jX] =2X0X1=b

0b1b2bkb

0V[b0] Cov[b0;b1] Cov[b0;b2]Cov[b0;bk]

b1Cov[ b0;b1]V[b1] Cov[b1;b2]Cov[b1;bk] b2Cov[ b0;b2] Cov[b1;b2]V[b2]Cov[b2;bk] bkCov[ b0;bk] Cov[bk;b1] Cov[bk;b2]V[bk] Recall that standard errors are the square root of the diagonals of this matrix. Stewart (Princeton)Week 7: Multiple RegressionOctober 22, 24, 2018 22 / 140

Overview of Inference in the General Setting

Under assumption 1-5 in large samples:

b jjc SE[bj]N(0;1)In small samples, under assumptions 1-6, b jjc

SE[bj]tn(k+1)Estimated standard errors are:

cquotesdbs_dbs14.pdfusesText_20

[PDF] Week 7: Multiple Regression i.e. analogous to the

How do you manually calculate multiple regression?

What is the formula for calculating multiple regression?

How do you manually calculate regression?

Week 7: Multiple Regression

Brandon Stewart

Princeton

October 22, 24, 20181

These slides are heavily in

Jens Hainmueller and Erin Hartman.

Where We've Been and Where We're Going...

Last Week

IMonday:

Wednesday:

Fagnostic regression

Fthe bootstrapNext Week

Ithen:::diagnosticsLong Run

Questions?

1Matrix Form of Regression

2OLS inference in matrix form

3Standard Hypothesis Tests

4Testing Joint Signicance

5Testing Linear Hypotheses: The General Case

6Fun With(out) Weights

7Appendix: Derivations and Consistency

8The Problems withp-values9Agnostic Regression

10Inference via the Bootstrap

11Fun With Weights

12Appendix: Trickyp-value ExampleStewart (Princeton)Week 7: Multiple RegressionOctober 22, 24, 2018 3 / 140

The Linear Model with New Notation

1=0+x11+z12+u1(unit 1)y

2=0+x21+z22+u2(unit 2)y

3=0+x31+z32+u3(unit 3)y

4=0+x41+z42+u4(unit 4)Stewart (Princeton)Week 7: Multiple RegressionOctober 22, 24, 2018 4 / 140

The Linear Model with New Notation

1=0+x11+z12+u1(unit 1)

2=0+x21+z22+u2(unit 2)

3=0+x31+z32+u3(unit 3)

4=0+x41+z42+u4(unit 4)We can write this as:

75Outcome is alinea rcombination of the the x,z, anduvectorsStewart (Princeton)Week 7: Multiple RegressionOctober 22, 24, 2018 5 / 140

Grouping Things into Matrices

Can we write this in a more compact form?

Let Xandbe the following:X

641x1z1

1x4z43

Back to Regression

Multiple Linear Regression in Matrix Form

0b1...

775=Xb=2

1b0+x21b1+x22b2++x2Kbk...

775Stewart (Princeton)Week 7: Multiple RegressionOctober 22, 24, 2018 8 / 140

Residuals

We can easily write the

OLS Estimator in Matrix Form

Goal: minimize the

Resulting rst order conditions:

0(yXb) = 0Rearranging:

0Xb=X0yIn order to isolate

Back to OLS

Intuition for the OLS in Matrix Form

1Matrix Form of Regression

2OLS inference in matrix form

3Standard Hypothesis Tests

4Testing Joint Signicance

5Testing Linear Hypotheses: The General Case

6Fun With(out) Weights

7Appendix: Derivations and Consistency

8The Problems withp-values9Agnostic Regression

10Inference via the Bootstrap

11Fun With Weights

12Appendix: Trickyp-value ExampleStewart (Princeton)Week 7: Multiple RegressionOctober 22, 24, 2018 13 / 140

OLS Assumptions in Matrix Form

Assumption 3: No Perfect Collinearity

Denition (Rank)

Assumption 5: Homoskedasticity

664var(u1) cov(u1;u2):::cov(u1;un)

775This matrix is alwayssymmetric since cov( ui;uj) = cov(uj;ui) by

Assumption 5: The Meaning of Homoskedasticity

What does var(ujX) =2uInmean?I

2u0 0:::0