4.1 — Multivariate OLS Estimators

ECON 480 • Econometrics • Fall 2022

Dr. Ryan Safner
Associate Professor of Economics

safner@hood.edu
ryansafner/metricsF22
metricsF22.classes.ryansafner.com

Contents

The Multivariate OLS Estimators

The Expected Value of ˆβj: Bias

Precision of ^βj

A Summary of Multivariate OLS Estimator Properties

(Updated) Measures of Fit

The Multivariate OLS Estimators

The Multivariate OLS Estimators

Yi=β0+β1X1i+β2X2i+⋯+βkXki+ui

  • The ordinary least squares (OLS) estimators of the unknown population parameters β0,β1,β2,⋯,βk solves:

min^β0,^β1,^β2,⋯,^βkn∑i=1[Yi−(^β0+^β1X1i+^β2X2i+⋯+^βkXki)⏟ˆYi⏟ˆui]2

  • Again, OLS estimators are chosen to minimize the sum of squared residuals (SSR)
    • i.e. sum of squared “distances” between actual values of Yi and predicted values ^Yi

The Multivariate OLS Estimators: FYI

Math FYI

in linear algebra terms, a regression model with n observations of k independent variables:

Y=Xβ+u

(y1y2⋮yn)⏟Y(n×1)=(x1,1x1,2⋯x1,nx2,1x2,2⋯x2,n⋮⋮⋱⋮xk,1xk,2⋯xk,n)⏟X(n×k)(β1β2⋮βk)⏟β(k×1)+(u1u2⋮un)⏟u(n×1)

  • The OLS estimator for β is ˆβ=(X′X)−1X′Y 😱

  • Appreciate that I am saving you from such sorrow 🤖

The Sampling Distribution of ^βj

  • For any individual βj, it has a sampling distribution:

^βj∼N(E[^βj],se(^βj))

  • We want to know its sampling distribution’s:
    • Center: E[^βj]; what is the expected value of our estimator?
    • Spread: se(^βj); how precise or uncertain is our estimator?

The Sampling Distribution of ^βj

  • For any individual βj, it has a sampling distribution:

^βj∼N(E[^βj],se(^βj))

  • We want to know its sampling distribution’s:
    • Center: E[^βj]; what is the expected value of our estimator?
    • Spread: se(^βj); how precise or uncertain is our estimator?

The Expected Value of ^βj: Bias

Exogeneity and Unbiasedness

  • As before, E[^βj]=βj when Xj is exogenous (i.e. cor(Xj,u)=0)
  • We know the true E[^βj]=βj+cor(Xj,u)σuσXj⏟O.V. Bias
  • If Xj is endogenous (i.e. cor(Xj,u)≠0), contains omitted variable bias
  • Let’s “see” an example of omitted variable bias and quantify it with our example

Measuring Omitted Variable Bias I

  • Suppose the true population model of a relationship is:

Yi=β0+β1X1i+β2X2i+ui

  • What happens when we run a regression and omit X2i?
  • Suppose we estimate the following omitted regression of just Yi on X1i (omitting X2i):1

Yi=α0+α1X1i+νi

  1. Note: I am using α’s and νi only to denote these are different estimates than the true model β’s and ui

Measuring Omitted Variable Bias II

  • Key Question: are X1i and X2i correlated?
  • Run an auxiliary regression of X2i on X1i to see:1

X2i=δ0+δ1X1i+τi

  • If δ1=0, then X1i and X2i are not linearly related

  • If |δ1| is very big, then X1i and X2i are strongly linearly related

  1. Note: I am using δ’s and τ to differentiate estimates for this model.

Measuring Omitted Variable Bias III

  • Now substitute our auxiliary regression between X2i and X1i into the true model:
    • We know X2i=δ0+δ1X1i+τi

Yi=β0+β1X1i+β2X2i+ui

Measuring Omitted Variable Bias III

  • Now substitute our auxiliary regression between X2i and X1i into the true model:
    • We know X2i=δ0+δ1X1i+τi

Yi=β0+β1X1i+β2X2i+uiYi=β0+β1X1i+β2(δ0+δ1X1i+τi)+ui

Measuring Omitted Variable Bias III

  • Now substitute our auxiliary regression between X2i and X1i into the true model:
    • We know X2i=δ0+δ1X1i+τi

Yi=β0+β1X1i+β2X2i+uiYi=β0+β1X1i+β2(δ0+δ1X1i+τi)+uiYi=(β0+β2δ0)+(β1+β2δ1)X1i+(β2τi+ui)

Measuring Omitted Variable Bias III

  • Now substitute our auxiliary regression between X2i and X1i into the true model:
    • We know X2i=δ0+δ1X1i+τi

Yi=β0+β1X1i+β2X2i+uiYi=β0+β1X1i+β2(δ0+δ1X1i+τi)+uiYi=(β0+β2δ0⏟α0)+(β1+β2δ1⏟α1)X1i+(β2τi+ui⏟νi)

  • Now relabel each of the three terms as the OLS estimates (α’s) and error (νi) from the omitted regression, so we again have:

Yi=α0+α1X1i+νi

  • Crucially, this means that our OLS estimate for X1i in the omitted regression is:

α1=β1+β2δ1

Measuring Omitted Variable Bias IV

α1=β1+β2δ1

  • The Omitted Regression OLS estimate for X1, (α1) picks up both:
  1. The true effect of X1 on Y: β1
  1. The true effect of X2 on Y: β2…as pulled through the relationship between X1 and X2: δ1
  • Recall our conditions for omitted variable bias from some variable Zi:
  1. Zi must be a determinant of Yi ⟹ β2≠0
  1. Zi must be correlated with Xi ⟹ δ1≠0
  • Otherwise, if Zi does not fit these conditions, α1=β1 and the omitted regression is unbiased!

Measuring OVB in Our Class Size Example I

  • The “True” Regression (Yi on X1i and X2i)

^Test Scorei=686.03−1.10 STRi−0.65 %ELi

ABCDEFGHIJ0123456789
term
<chr>
estimate
<dbl>
std.error
<dbl>
statistic
<dbl>
p.value
<dbl>
(Intercept)686.03224877.4113124892.5655543.871501e-280
str-1.10129590.38027832-2.8960263.978056e-03
el_pct-0.64977680.03934255-16.5158791.657506e-47
3 rows

Measuring OVB in Our Class Size Example II

  • The “Omitted” Regression (Yi on just X1i)

^Test Scorei=698.93−2.28 STRi

ABCDEFGHIJ0123456789
term
<chr>
estimate
<dbl>
std.error
<dbl>
statistic
<dbl>
p.value
<dbl>
(Intercept)698.9329529.467491473.8245146.569925e-242
str-2.2798080.4798256-4.7513272.783307e-06
2 rows

Measuring OVB in Our Class Size Example III

  • The “Auxiliary” Regression (X2i on X1i)

^%ELi=−19.85+1.81 STRi

ABCDEFGHIJ0123456789
term
<chr>
estimate
<dbl>
std.error
<dbl>
statistic
<dbl>
p.value
<dbl>
(Intercept)-19.8540559.1626044-2.1668570.0308099863
str1.8137190.46437353.9057330.0001095165
2 rows

Measuring OVB in Our Class Size Example IV

“True” Regression

^Test Scorei=686.03−1.10 STRi−0.65 %EL

“Omitted” Regression

^Test Scorei=698.93−2.28 STRi

“Auxiliary” Regression

^%ELi=−19.85+1.81 STRi

  • Omitted Regression α1 on STR is -2.28

Measuring OVB in Our Class Size Example IV

“True” Regression

^Test Scorei=686.03−1.10 STRi−0.65 %EL

“Omitted” Regression

^Test Scorei=698.93−2.28 STRi

“Auxiliary” Regression

^%ELi=−19.85+1.81 STRi

  • Omitted Regression α1 on STR is -2.28

α1=β1+β2δ1

  • The true effect of STR on Test Score: -1.10

Measuring OVB in Our Class Size Example IV

“True” Regression

^Test Scorei=686.03−1.10 STRi−0.65 %EL

“Omitted” Regression

^Test Scorei=698.93−2.28 STRi

“Auxiliary” Regression

^%ELi=−19.85+1.81 STRi

  • Omitted Regression α1 on STR is -2.28

α1=β1+β2δ1

  • The true effect of STR on Test Score: -1.10

  • The true effect of %EL on Test Score: -0.65

Measuring OVB in Our Class Size Example IV

“True” Regression

^Test Scorei=686.03−1.10 STRi−0.65 %EL

“Omitted” Regression

^Test Scorei=698.93−2.28 STRi

“Auxiliary” Regression

^%ELi=−19.85+1.81 STRi

  • Omitted Regression α1 on STR is -2.28

α1=β1+β2δ1

  • The true effect of STR on Test Score: -1.10

  • The true effect of %EL on Test Score: -0.65

  • The relationship between STR and %EL: 1.81

Measuring OVB in Our Class Size Example IV

“True” Regression

^Test Scorei=686.03−1.10 STRi−0.65 %EL

“Omitted” Regression

^Test Scorei=698.93−2.28 STRi

“Auxiliary” Regression

^%ELi=−19.85+1.81 STRi

  • Omitted Regression α1 on STR is -2.28

α1=β1+β2δ1

  • The true effect of STR on Test Score: -1.10

  • The true effect of %EL on Test Score: -0.65

  • The relationship between STR and %EL: 1.81

  • So, for the omitted regression:

−2.28=−1.10+(−0.65)(1.81)

Measuring OVB in Our Class Size Example IV

“True” Regression

^Test Scorei=686.03−1.10 STRi−0.65 %EL

“Omitted” Regression

^Test Scorei=698.93−2.28 STRi

“Auxiliary” Regression

^%ELi=−19.85+1.81 STRi

  • Omitted Regression α1 on STR is -2.28

α1=β1+β2δ1

  • The true effect of STR on Test Score: -1.10

  • The true effect of %EL on Test Score: -0.65

  • The relationship between STR and %EL: 1.81

  • So, for the omitted regression:

−2.28=−1.10+(−0.65)(1.81)⏟O.V.Bias=−1.18

Precision of ^βj

Precision of ^βj I

  • σ^βj; how precise or uncertain are our estimates?

  • Variance σ2^βj or standard error σ^βj

Precision of ^βj II

var(^βj)=11−R2j⏟VIF×(SER)2n×var(X)

se(^βj)=√var(^βj)

  • Variation in ^βj is affected by four things now1:
  1. Goodness of fit of the model (SER)
    • Larger SER → larger var(^βj)
  2. Sample size, n
    • Larger n → smaller var(^βj)
  3. Variance of X
    • Larger var(X) → smaller var(^βj)
  4. Variance Inflation Factor 1(1−R2j)
    • Larger VIF, larger var(^βj)
    • This is the only new effect

  1. See Class 2.5 for a reminder of variation with just one X variable.

VIF and Multicollinearity I

  • Two independent (X) variables are multicollinear:

cor(Xj,Xl)≠0∀j≠l

  • Multicollinearity between X variables does not bias OLS estimates
    • Remember, we pulled another variable out of u into the regression
    • If it were omitted, then it would cause omitted variable bias!
  • Multicollinearity does increase the variance of each OLS estimator by

VIF=1(1−R2j)

VIF and Multicollinearity II

VIF=1(1−R2j)

  • R2j is the R2 from an auxiliary regression of Xj on all other regressors (X’s)
    • i.e. proportion of var(Xj) explained by other X’s

VIF and Multicollinearity III

Example

Suppose we have a regression with three regressors (k=3):

Yi=β0+β1X1i+β2X2i+β3X3i+ui

  • There will be three different R2j’s, one for each regressor:

R21 for X1i=γ+γX2i+γX3iR22 for X2i=ζ0+ζ1X1i+ζ2X3iR23 for X3i=η0+η1X1i+η2X2i

VIF and Multicollinearity IV

VIF=1(1−R2j)

  • R2j is the R2 from an auxiliary regression of Xj on all other regressors (X’s)

    • i.e. proportion of var(Xj) explained by other X’s

  • The R2j tells us how much other regressors explain regressor Xj

  • Key Takeaway: If other X variables explain Xj well (high R2J), it will be harder to tell how cleanly Xj→Yi, and so var(^βj) will be higher

VIF and Multicollinearity V

  • Common to calculate the Variance Inflation Factor (VIF) for each regressor:

VIF=1(1−R2j)

  • VIF quantifies the factor (scalar) by which var(^βj) increases because of multicollinearity
    • e.g. VIF of 2, 3, etc. ⟹ variance increases by 2x, 3x, etc.
  • Baseline: R2j=0 ⟹ no multicollinearity ⟹VIF=1 (no inflation)
  • Larger R2j ⟹ larger VIF
    • Rule of thumb: VIF>10 is problematic

VIF and Multicollinearity in Our Example I

  • Higher %EL predicts higher STR
  • Hard to get a precise marginal effect of STR holding %EL constant
    • Don’t have much data on districts with low STR and high %EL (and vice versa)!

VIF and Multicollinearity in Our Example II

  • Again, consider the correlation between the variables
ca_school %>%
  # Select only the three variables we want (there are many)
  select(str, testscr, el_pct) %>%
  # make a correlation table (all variables must be numeric)
  cor()
               str    testscr     el_pct
str      1.0000000 -0.2263628  0.1876424
testscr -0.2263628  1.0000000 -0.6441237
el_pct   0.1876424 -0.6441237  1.0000000
  • cor(STR,%EL)=−0.644

VIF and Multicollinearity in R I

# our multivariate regression
elreg <- lm(testscr ~ str + el_pct,
            data = ca_school)

# use the "car" package for VIF function 
library("car")

elreg %>% vif()
     str   el_pct 
1.036495 1.036495
  • var(^β1) on str increases by 1.036 times (3.6%) due to multicollinearity with el_pct
  • var(^β2) on el_pct increases by 1.036 times (3.6%) due to multicollinearity with str

VIF and Multicollinearity in R II

  • Let’s calculate VIF manually to see where it comes from:
# run auxiliary regression of x2 on x1
auxreg <- lm(el_pct ~ str,
             data = ca_school)

library(broom)
auxreg %>% tidy() # look at reg output
ABCDEFGHIJ0123456789
term
<chr>
estimate
<dbl>
std.error
<dbl>
statistic
<dbl>
p.value
<dbl>
(Intercept)-19.8540559.1626044-2.1668570.0308099863
str1.8137190.46437353.9057330.0001095165
2 rows

VIF and Multicollinearity in R III

auxreg %>% glance() # look at aux reg stats for R^2
ABCDEFGHIJ0123456789
r.squared
<dbl>
adj.r.squared
<dbl>
sigma
<dbl>
statistic
<dbl>
p.value
<dbl>
0.035209660.0329015517.9825915.254750.0001095165
1 row | 1-5 of 12 columns
# extract our R-squared from aux regression (R_j^2)

aux_r_sq <- glance(auxreg) %>%
  pull(r.squared)

aux_r_sq # look at it
[1] 0.03520966

VIF and Multicollinearity in R IV

# calculate VIF manually

our_vif <- 1 / (1 - aux_r_sq) # VIF formula 

our_vif
[1] 1.036495
  • Again, multicollinearity between the two X variables inflates the variance on each by 1.036 times

Another Example: Expenditures/Student I

Example

What about district expenditures per student?

ca_school %>%
  select(testscr, str, el_pct, expn_stu) %>%
  cor()
            testscr        str      el_pct    expn_stu
testscr   1.0000000 -0.2263628 -0.64412374  0.19127277
str      -0.2263628  1.0000000  0.18764237 -0.61998215
el_pct   -0.6441237  0.1876424  1.00000000 -0.07139604
expn_stu  0.1912728 -0.6199821 -0.07139604  1.00000000

Another Example: Expenditures/Student II

  • Higher spend predicts lower STR
  • Hard to get a precise marginal effect of STR holding spend constant
    • Don’t have much data on districts with high STR and high spend (and vice versa)!

Another Example: Expenditures/Student II

Would omitting Expenditures per student cause omitted variable bias?

  1. cor(Test,spend)≠0

  2. cor(STR,spend)≠0

Another Example: Expenditures/Student III

ABCDEFGHIJ0123456789
term
<chr>
estimate
<dbl>
std.error
<dbl>
statistic
<dbl>
p.value
<dbl>
(Intercept)649.57794725715.20571855442.71931943.299149e-154
str-0.2863992400.480523198-0.59601545.514891e-01
el_pct-0.6560226600.039105885-16.77554811.295346e-48
expn_stu0.0038679020.0014121222.73906986.426186e-03
4 rows
vif(reg3)
     str   el_pct expn_stu 
1.680787 1.040031 1.629915
  • Including expn_stu reduces bias but increases variance of β1 by 1.68x (68%)
    • and variance of β2 by 1.04x (4%)

Multicollinearity Increases Variance

Test Scores Test Scores Test Scores
Constant 698.93*** 686.03*** 649.58***
(9.47) (7.41) (15.21)
Student Teacher Ratio −2.28*** −1.10*** −0.29
(0.48) (0.38) (0.48)
Percent ESL Students −0.65*** −0.66***
(0.04) (0.04)
Spending per Student 0.00***
(0.00)
n 420 420 420
R2 0.05 0.43 0.44
SER 18.54 14.41 14.28
* p < 0.1, ** p < 0.05, *** p < 0.01

Perfect Multicollinearity

  • Perfect multicollinearity is when a regressor is an exact linear function of (an)other regressor(s)

^Sales=^β0+^β1Temperature (C)+^β2Temperature (F)

Temperature (F)=32+1.8∗Temperature (C)

  • cor(temperature (F), temperature (C))=1
  • R2j=1→VIF=11−1→var(^βj)=0!
  • This is fatal for a regression
    • A logical impossiblity, always caused by human error

Perfect Multicollinearity: Example

Example

^TestScorei=^β0+^β1STRi+^β2%EL+^β3%EF

  • %EL: the percentage of students learning English

  • %EF: the percentage of students fluent in English

  • %EF=100−%EL

  • |cor(%EF,%EL)|=1

Perfect Multicollinearity: Example II

# generate %EF variable from %EL
ca_school_ex <- ca_school %>%
  mutate(ef_pct = 100 - el_pct)

# get correlation between %EL and %EF
ca_school_ex %>%
  summarize(cor = cor(ef_pct, el_pct))
ABCDEFGHIJ0123456789
cor
<dbl>
-1
1 row

Perfect Multicollinearity: Example III

Perfect Multicollinearity Example IV

mcreg <- lm(testscr ~ str + el_pct + ef_pct,
            data = ca_school_ex)
summary(mcreg)

Call:
lm(formula = testscr ~ str + el_pct + ef_pct, data = ca_school_ex)

Residuals:
    Min      1Q  Median      3Q     Max 
-48.845 -10.240  -0.308   9.815  43.461 

Coefficients: (1 not defined because of singularities)
             Estimate Std. Error t value Pr(>|t|)    
(Intercept) 686.03225    7.41131  92.566  < 2e-16 ***
str          -1.10130    0.38028  -2.896  0.00398 ** 
el_pct       -0.64978    0.03934 -16.516  < 2e-16 ***
ef_pct             NA         NA      NA       NA    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 14.46 on 417 degrees of freedom
Multiple R-squared:  0.4264,    Adjusted R-squared:  0.4237 
F-statistic:   155 on 2 and 417 DF,  p-value: < 2.2e-16
mcreg %>% tidy()
ABCDEFGHIJ0123456789
term
<chr>
estimate
<dbl>
std.error
<dbl>
statistic
<dbl>
p.value
<dbl>
(Intercept)686.03224877.4113124892.5655543.871501e-280
str-1.10129590.38027832-2.8960263.978056e-03
el_pct-0.64977680.03934255-16.5158791.657506e-47
ef_pctNANANANA
4 rows
  • Note R drops one of the multicollinear regressors (ef_pct) if you include both 🤡

A Summary of Multivariate OLS Estimator Properties

A Summary of Multivariate OLS Estimator Properties

  • ^βj on Xj is biased only if there is an omitted variable (Z) such that:
    1. cor(Y,Z)≠0
    2. cor(Xj,Z)≠0
    • If Z is included and Xj is collinear with Z, this does not cause a bias
  • var[^βj] and se[^βj] measure precision (or uncertainty) of estimate:

var[^βj]=1(1−R2j)∗SER2n×var[Xj]

  • VIF from multicollinearity: 1(1−R2j)
    • R2j for auxiliary regression of Xj on all other X’s
    • mutlicollinearity does not bias ^βj but raises its variance
    • perfect multicollinearity if X’s are linear function of others

(Updated) Measures of Fit

(Updated) Measures of Fit

  • Again, how well does a linear model fit the data?

  • How much variation in Yi is “explained” by variation in the model (^Yi)?

Yi=^Yi+^ui^ui=Yi−^Yi

(Updated) Measures of Fit: SER

  • Again, the Standard errror of the regression (SER) estimates the standard error of u

SER=SSRn−k−1

  • A measure of the spread of the observations around the regression line (in units of Y), the average “size” of the residual

  • Only new change: divided by n−k−1 due to use of k+1 degrees of freedom to first estimate β0 and then all of the other β’s for the k number of regressors1

  1. Again, because your textbook defines k as including the constant, the denominator would be n-k instead of n-k-1.

(Updated) Measures of Fit: R2

R2=SSMSST=1−SSRSST=(rX,Y)2

  • Again, R2 is fraction of total variation in Yi (“total sum of squares”) that is explained by variation in predicted values (^Yi), i.e. our model (“model sum of squares”)

R2=var(ˆY)var(Y)

Visualizing R2

  • Total Variation in Y: Areas A + D + E + G

SST=n∑i=1(Yi−ˉY)2

  • Variation in Y explained by X1 and X2: Areas D + E + G

SSM=n∑i=1(^Yi−ˉY)2

  • Unexplained variation in Y: Area A

SSR=n∑i=1(^ui)2

Compare with one X variable

R2=SSMSST=D+E+GA+D+E+G

Visualizing R2

# make a function to calc. sum of sq. devs
sum_sq <- function(x){sum((x - mean(x))^2)}

# find total sum of squares
SST <- elreg %>%
  augment() %>%
  summarize(SST = sum_sq(testscr))

# find explained sum of squares
SSM <- elreg %>%
  augment() %>%
  summarize(SSM = sum_sq(.fitted))

# look at them and divide to get R^2
tribble(
  ~SSM, ~SST, ~R_sq,
  SSM, SST, SSM/SST
  ) %>%
  knitr::kable()
SSM SST R_sq
64864.3 152109.6 0.4264314

R2=SSMSST=D+E+GA+D+E+G

(Updated) Measures of Fit: Adjusted ˉR2

  • Problem: R2 mechanically increases every time a new variable is added (it reduces SSR!)
    • Think in the diagram: more area of Y covered by more X variables!
  • This does not mean adding a variable improves the fit of the model per se, R2 gets inflated
  • We correct for this effect with the adjusted ˉR2 which penalizes adding new variables:

ˉR2=1−n−1n−k−1⏟penalty×SSRSST

  • In the end, recall R2 was never that useful1, so don’t worry about the formula
    • Large sample sizes (n) make R2 and ˉR2 very close

  1. …for measuring causal effects (our goal). It is useful if you care about prediction instead!

ˉR2 In R

summary(elreg)

Call:
lm(formula = testscr ~ str + el_pct, data = ca_school)

Residuals:
    Min      1Q  Median      3Q     Max 
-48.845 -10.240  -0.308   9.815  43.461 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept) 686.03225    7.41131  92.566  < 2e-16 ***
str          -1.10130    0.38028  -2.896  0.00398 ** 
el_pct       -0.64978    0.03934 -16.516  < 2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 14.46 on 417 degrees of freedom
Multiple R-squared:  0.4264,    Adjusted R-squared:  0.4237 
F-statistic:   155 on 2 and 417 DF,  p-value: < 2.2e-16
glance(elreg)
ABCDEFGHIJ0123456789
r.squared
<dbl>
adj.r.squared
<dbl>
sigma
<dbl>
statistic
<dbl>
p.value
<dbl>
0.42643140.423680414.46448155.01364.624399e-51
1 row | 1-5 of 12 columns
  • Base R2 (R calls it “Multiple R-squared”) went up
  • Adjusted R-squared (ˉR2) went down

Coefficient Plots (with modelsummary)

  • Plot
  • Code

library(modelsummary)
modelplot(reg3,  # our regression object
          coef_omit = 'Intercept') # don't show intercept

Regression Table (with modelsummary)

  • Output
  • Code
Simple Model MV Model 1 MV Model 2
Constant 698.93*** 686.03*** 649.58***
(9.47) (7.41) (15.21)
STR −2.28*** −1.10*** −0.29
(0.48) (0.38) (0.48)
% ESL Students −0.65*** −0.66***
(0.04) (0.04)
Spending per Student 0.00***
(0.00)
N 420 420 420
Adj. R2 0.05 0.42 0.43
SER 18.54 14.41 14.28
* p < 0.1, ** p < 0.05, *** p < 0.01
 
modelsummary(models = list("Simple Model" = school_reg,
                           "MV Model 1" = elreg,
                           "MV Model 2" = reg3),
             fmt = 2, # round to 2 decimals
             output = "html",
             coef_rename = c("(Intercept)" = "Constant",
                             "str" = "STR",
                             "el_pct" = "% ESL Students",
                             "expn_stu" = "Spending per Student"),
             gof_map = list(
               list("raw" = "nobs", "clean" = "N", "fmt" = 0),
               #list("raw" = "r.squared", "clean" = "R<sup>2</sup>", "fmt" = 2),
               list("raw" = "adj.r.squared", "clean" = "Adj. R<sup>2</sup>", "fmt" = 2),
               list("raw" = "rmse", "clean" = "SER", "fmt" = 2)
             ),
             escape = FALSE,
             stars = c('*' = .1, '**' = .05, '***' = 0.01)
)

ECON 480 — Econometrics

1
4.1 — Multivariate OLS Estimators ECON 480 • Econometrics • Fall 2022 Dr. Ryan Safner Associate Professor of Economics safner@hood.edu ryansafner/metricsF22 metricsF22.classes.ryansafner.com

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  • Title Slide
  • Contents
  • The Multivariate OLS Estimators
  • The Multivariate OLS Estimators
  • The Multivariate OLS Estimators: FYI
  • The Sampling Distribution of \(\hat{\beta_j}\)
  • The Sampling Distribution of \(\hat{\beta_j}\)
  • The Expected Value of \(\hat{\beta_j}\): Bias
  • Exogeneity and Unbiasedness
  • Measuring Omitted Variable Bias I
  • Measuring Omitted Variable Bias II
  • Measuring Omitted Variable Bias III
  • Measuring Omitted Variable Bias III
  • Measuring Omitted Variable Bias III
  • Measuring Omitted Variable Bias III
  • Measuring Omitted Variable Bias IV
  • Measuring OVB in Our Class Size Example I
  • Measuring OVB in Our Class Size Example II
  • Measuring OVB in Our Class Size Example III
  • Measuring OVB in Our Class Size Example IV
  • Measuring OVB in Our Class Size Example IV
  • Measuring OVB in Our Class Size Example IV
  • Measuring OVB in Our Class Size Example IV
  • Measuring OVB in Our Class Size Example IV
  • Measuring OVB in Our Class Size Example IV
  • Precision of \(\hat{\beta_j}\)
  • Precision of \(\hat{\beta_j}\) I
  • Precision of \(\hat{\beta_j}\) II
  • VIF and Multicollinearity I
  • VIF and Multicollinearity II
  • VIF and Multicollinearity III
  • VIF and Multicollinearity IV
  • VIF and Multicollinearity V
  • VIF and Multicollinearity in Our Example I
  • VIF and Multicollinearity in Our Example II
  • VIF and Multicollinearity in R I
  • VIF and Multicollinearity in R II
  • VIF and Multicollinearity in R III
  • VIF and Multicollinearity in R IV
  • Another Example: Expenditures/Student I
  • Another Example: Expenditures/Student II
  • Another Example: Expenditures/Student II
  • Another Example: Expenditures/Student III
  • Multicollinearity Increases Variance
  • Perfect Multicollinearity
  • Perfect Multicollinearity: Example
  • Perfect Multicollinearity: Example II
  • Perfect Multicollinearity: Example III
  • Perfect Multicollinearity Example IV
  • A Summary of Multivariate OLS Estimator Properties
  • A Summary of Multivariate OLS Estimator Properties
  • (Updated) Measures of Fit
  • (Updated) Measures of Fit
  • (Updated) Measures of Fit: SER
  • (Updated) Measures of Fit: \(R^2\)
  • Visualizing \(R^2\)
  • Visualizing \(R^2\)
  • (Updated) Measures of Fit: Adjusted \(\bar{R}^2\)
  • \(\bar{R}^2\) In R
  • Coefficient Plots (with modelsummary)
  • Regression Table (with modelsummary)
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