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Regression Analysis
• Linear Regression Model
– Method of OLS
– Properties of OLS Estimators
– Goodness-of-Fit
– Inference
• Multiple Regression Model
– Estimation
– Goodness-of-Fit
– Inference
Sisir Sarma 18.318: Introduction to Econometrics
The Simple Regression Model
• Economic Model: y = b0 + b1x
• Examples: Consumption Function, Savings
Function, Demand Function, Supply Function, etc.
• The parameters we are interested in this model
are: b0 and b1, which we wish to estimate.
• A simple regression model can be written as
y = b0 + b1 x +
Sisir Sarma 18.318: Introduction to Econometrics
Some Terminology
In the simple linear regression model, where y =
b0 + b1x + , we typically refer to y as the
• Dependent Variable, or
• Left-Hand Side Variable, or
• Explained Variable, or
• Regressand
Sisir Sarma 18.318: Introduction to Econometrics
Some Terminology (cont.)
In the simple linear regression of y on x, we
typically refer to x as the
• Independent Variable, or
• Right-Hand Side Variable, or
• Explanatory Variable, or
• Regressor, or
• Covariate, or
• Control Variable
Sisir Sarma 18.318: Introduction to Econometrics
A Simple Assumption
The average value of , the error term, in the
population is 0. That is,
E() = 0
This is not a restrictive assumption, since we can
always use b0 to normalize E() to 0.
Sisir Sarma 18.318: Introduction to Econometrics
Zero Conditional Mean
• We need to make a crucial assumption about
how and x are related
• We want it to be the case that knowing
something about x does not give us any
information about , so that they are
completely unrelated. That is,
• E(|x) = E() = 0, which implies
• E(y|x) = b0 + b1x
Sisir Sarma 18.318: Introduction to Econometrics
Ordinary Least Squares
• Basic idea of regression is to estimate the
population parameters from a sample.
• Let {(xi,yi): i = 1, …, n} denote a random
sample of size n from the population.
• For each observation in this sample, it will be
the case that
yi = b0 + b1xi + i
This is the econometric model.
Sisir Sarma 18.318: Introduction to Econometrics
Population regression line, sample data points
and the associated error terms
y E(y|x) = b0 + b1x
y4 .
4 {
y3 .} 3
y2 2 {
.
1
y1 .}
x1 x2 x3 x4 x
Deriving OLS Estimates
• To derive the OLS estimates we need to realize
that our main assumption of E(|x) = E( ) = 0
also implies that
• Cov(x, ) = E(x ) = 0
• Why? Remember from basic probability that
Cov(X,Y) = E(XY) – E(X)E(Y).
Sisir Sarma 18.318: Introduction to Econometrics
Deriving OLS (cont.)
• We can write our 2 restrictions just in terms of
x, y, b0 and b1 , since = y – b0 – b1x
• E(y – b0 – b1x) = 0
• E[x(y – b0 – b1x)] = 0
• These are called moment restrictions
Sisir Sarma 18.318: Introduction to Econometrics
Deriving OLS using M.O.M.
• The method of moments approach to estimation
implies imposing the population moment
restrictions on the sample moments.
• What does this mean? Recall that for E(X), the
mean of a population distribution, a sample
estimator of E(X) is simply the arithmetic mean
of the sample.
Sisir Sarma 18.318: Introduction to Econometrics
More Derivation of OLS
• We want to choose values of the parameters
that will ensure that the sample versions of our
moment restrictions are true.
• The sample versions are as follows:
n
1 ˆ ˆ
n i 1
y i b 0 b1 xi 0
n
1 ˆ ˆ
n i 1
xi y i b 0 b 1 xi 0
Sisir Sarma 18.318: Introduction to Econometrics
More Derivation of OLS
• Given the definition of a sample mean, and
properties of summation, we can rewrite the
first condition as follows
ˆ ˆ
y b 0 b1 x ,
or
ˆ
ˆ yb x
b0 1
Sisir Sarma 18.318: Introduction to Econometrics
The OLS estimated slope is
n
x x y
i i y
ˆ
b1 i 1
n
x x
2
i
i 1
n
provided that xi x 0
2
i 1
Sisir Sarma 18.318: Introduction to Econometrics
Summary of OLS slope estimate
• The slope estimate is the sample covariance between x
and y divided by the sample variance of x. [Note: if
you divide both the numerator and the denominator by
(n-1), we get the sample covariance and the sample
variance formulas, respectively].
• If x and y are positively correlated, the slope will be
positive.
• If x and y are negatively correlated, the slope will be
negative.
• Only need x to vary in our sample.
Sisir Sarma 18.318: Introduction to Econometrics
More OLS
• Intuitively, OLS is fitting a line through the
sample points such that the sum of squared
residuals is as small as possible, hence the term
least squares.
• The residual, is an estimate of the error
term, , and is the difference between the fitted
line (sample regression function) and the
sample point.
Sisir Sarma 18.318: Introduction to Econometrics
Sample regression line, sample data points
and the associated estimated error terms
y
y4 .
4{
ˆ ˆ ˆ
y b0 b1x
y3 .} 3
y2 .
2{
y1 .} 1
x1 x2 x3 x4 x
Alternate approach to Derivation
(The Textbook)
• Given the intuitive idea of fitting a line, we can
set up a formal minimization problem.
• That is, we want to choose our parameters such
that we minimize the SSR:
ˆi yi b 0 b1 x
n n
2 ˆ ˆ 2
i 1 i 1
Sisir Sarma 18.318: Introduction to Econometrics
Alternate approach (cont.)
• If one uses calculus to solve the minimization
problem for the two parameters you obtain the
following first order conditions, which are the
same as we obtained before, multiplied by n
y 0
n
i b 0 b 1 xi
ˆ ˆ
i 1
0
n
xi yi b 0 b1 xi
ˆ ˆ
i 1
Sisir Sarma 18.318: Introduction to Econometrics
Algebraic Properties of OLS
• The sum of the OLS residuals is zero.
• Thus, the sample average of the OLS residuals
is zero as well.
• The sample covariance between the regressors
and the OLS residuals is zero.
• The OLS regression line always goes through
the mean of the sample.
Sisir Sarma 18.318: Introduction to Econometrics
Algebraic Properties (precise)
n
n ˆi
ˆi 0 and thus, i 1
n
0
i 1
n
xiˆi 0
i 1
ˆ ˆ
y b 0 b1 x
Sisir Sarma 18.318: Introduction to Econometrics
More terminology
We can think of each observatio n as being made
up of an explained part, and an unexplaine d part,
y i y i i We then define the following :
ˆ ˆ
n
y i y 2 is the total sum of squares (TSS)
i 1
n
yi y
2
ˆ is the explained sum of squares (ESS)
i 1
n
i 2 is the sum of squared residuals
ˆ (SSR)
i 1
Then TSS ESS SSR OR ESS TSS - SSR
Sisir Sarma 18.318: Introduction to Econometrics
Proof that TSS = ESS + SSR
yi y yi yi yi y
2 2
ˆ ˆ
i y i y
2
ˆ ˆ
i 2 i y i y y i y
2
ˆ 2
ˆ ˆ ˆ
SSR 2 i y i y ESS
ˆ ˆ
and we know that i y i y 0
ˆ ˆ
Sisir Sarma 18.318: Introduction to Econometrics
Goodness-of-Fit
• How do we think about how well our sample
regression line fits our sample data?
• Can compute the fraction of the total sum of
squares (TSS) that is explained by the model,
call this the R-squared of regression
• R2 = ESS/TSS = 1 – SSR/TSS
• Since SSR lies between 0 and TSS, R2 will
always lie between 0 and 1.
Sisir Sarma 18.318: Introduction to Econometrics
Unbiasedness of OLS
• Assume the population model is linear in
parameters as y = b0 + b1x +
• Assume we can use a random sample of size n,
{(xi, yi): i = 1, 2, …, n}, from the population
model. Thus we can write the sample model yi
= b0 + b1xi + i
• Assume E(|x) = 0 and thus E(i|xi) = 0
• Assume there is variation in the xi
Sisir Sarma 18.318: Introduction to Econometrics
Unbiasedness of OLS (cont.)
• In order to think about unbiasedness, we need
to rewrite our estimator in terms of the
population parameter.
• Start with a simple rewrite of the formula as
ˆ
b1
x x yi i
x x
2
i
Sisir Sarma 18.318: Introduction to Econometrics
Unbiasedness of OLS (cont.)
ˆ b xi x i
b1
xi x
1 2
ˆ b
So, E b1 1
Sisir Sarma 18.318: Introduction to Econometrics
Unbiasedness Summary
• The OLS estimates of b1 and b0 are unbiased
• Proof of unbiasedness depends on our 4
assumptions – if any assumption fails, then
OLS is not necessarily unbiased
• Remember unbiasedness is a description of the
estimator – in a given sample we may be “near”
or “far” from the true parameter
Sisir Sarma 18.318: Introduction to Econometrics
Variance of the OLS Estimators
• Now we know that the sampling distribution of
our estimate is centered around the true
parameter.
• Want to think about how spread out this
distribution is.
• Much easier to think about this variance under
an additional assumption, so
• Assume Var(|x) = s2 (Homoskedasticity)
Sisir Sarma 18.318: Introduction to Econometrics
Variance of OLS (cont.)
• Var(|x) = E(2|x)-[E(|x)]2
• E(|x) = 0, so s 2 = E(2|x) = E(2) = Var()
• Thus s2 is also the unconditional variance,
called the error variance.
• s, the square root of the error variance is called
the standard deviation of the error.
• Can say: E(y|x)=b0 + b1x and Var(y|x) = s2.
Sisir Sarma 18.318: Introduction to Econometrics
Homoskedastic Case
y
f(y|x)
. E(y|x) = b + b x
0 1
.
x1 x2
Heteroskedastic Case
f(y|x)
.
. E(y|x) = b0 + b1x
.
x1 x2 x3 x
Variance of OLS
s 2
ˆ
Var b1
x x
2
i
s x
N x x
2 2
ˆ
Var b i
0 2
i
Sisir Sarma 18.318: Introduction to Econometrics
Variance of OLS Summary
• The larger the error variance, s2, the larger the
variance of the slope estimate, for a given x x 2
i
•
• The larger the variability in the xi, the smaller
the variance of the slope estimate.
• As a result, a larger sample size should
decrease the variance of the slope estimate.
• Problem that the error variance is unknown.
Sisir Sarma 18.318: Introduction to Econometrics
Estimating the Error Variance
• We don’t know what the error variance, s2 is,
because we don’t observe the errors, i.
• What we observe are the residuals, i
ˆ
• We can use the residuals to form an estimate of
the error variance.
Sisir Sarma 18.318: Introduction to Econometrics
Error Variance Estimate (cont.)
An unbiased estimator of s is 2
ˆi SSR /n 2
1
s
ˆ 2 2
n 2
Sisir Sarma 18.318: Introduction to Econometrics
Error Variance Estimate (cont.)
sˆ s 2 Standard error of the regression
ˆ
recall that sd b s
ˆ
xi x
1
2 2
if wesubstitutes for s then wehave
ˆ
ˆ
the standard error of b1 ,
ˆ
se b1 s / xi x
ˆ 2
1
2
Sisir Sarma 18.318: Introduction to Econometrics
Multiple Regression Analysis
y = b0 + b1x1 + b2x2 + . . . bkxk +
Estimation
Sisir Sarma 18.318: Introduction to Econometrics
Parallels with Simple Regression
• b0 is still the intercept
• b1 to bk all called slope parameters
• is still the error term
• Still need to make a zero conditional mean
assumption, so now assume that
• E(|x1,x2, …,xk) = 0
• Still minimizing the sum of squared
residuals, so have k+1 first order conditions
Sisir Sarma 18.318: Introduction to Econometrics
Interpreting Multiple Regression
ˆ ˆ ˆ ˆ ˆ
y b 0 b1 x1 b 2 x2 ... b k xk , so
ˆ ˆ ˆ
y b x b x ... b x , ˆ
1 1 2 2 k k
so holding x2 ,...,xk fixed implies that
ˆ
y b x , that is each b has
ˆ 1 1
a ceteris paribus interpretation
Sisir Sarma 18.318: Introduction to Econometrics
Simple vs Multiple Reg Estimate
~ ~
~b b x
Compare the simple regression y 0 1 1
ˆ ˆ ˆ ˆ
with the multiple regression y b 0 b1 x1 b 2 x2
~ ˆ
Generally, b1 b1 unless :
ˆ
b 0 (i.e. no partial effect of x ) OR
2 2
x1 and x2 are uncorrelated in the sample
Sisir Sarma 18.318: Introduction to Econometrics
Goodness-of-Fit
We can think of each observatio n as being made
up of an explained part, and an unexplaine d part,
yi yi i We then define the following :
ˆ ˆ
yi y is the total sum of squares (TSS)
2
yi y is the explained sum of squares (ESS)
2
ˆ
ˆi is the sum of squared residuals (SSR)
2
Then TSS ESS SSR
Sisir Sarma 18.318: Introduction to Econometrics
Goodness-of-Fit (cont.)
• How do we think about how well our
sample regression line fits our sample data?
• Can compute the fraction of the total sum
of squares (SST) that is explained by the
model, call this the R-squared of regression
• R2 = ESS/TSS = 1 – SSR/TSS
Sisir Sarma 18.318: Introduction to Econometrics
More about R-squared
• R2 can never decrease when another
independent variable is added to a
regression, and usually will increase
• Because R2 will usually increase with the
number of independent variables, it is not a
good way to compare models
Sisir Sarma 18.318: Introduction to Econometrics
Adjusted R-Squared
• Recall that the R2 will always increase as more
variables are added to the model
• The adjusted R2 takes into account the number of
variables in a model, and may decrease
R 2
1
SSR n k 1
SST n 1
sˆ 2
1
SST n 1
Sisir Sarma 18.318: Introduction to Econometrics
Adjusted R-Squared (cont.)
• Most packages will give you both R2 and
adj-R2
• You can compare the fit of 2 models (with
the same y) by comparing the adj-R2
• You cannot use the adj-R2 to compare
models with different y’s (e.g. y vs. ln(y))
Sisir Sarma 18.318: Introduction to Econometrics
Goodness of Fit
• Important not to fixate too much on adj-R2 and
lose sight of theory and common sense
• If economic theory clearly predicts a variable
belongs, generally leave it in
• Don’t want to include a variable that prohibits a
sensible interpretation of the variable of interest
• Remember ceteris paribus interpretation of
multiple regression
Sisir Sarma 18.318: Introduction to Econometrics
Classical Linear Model: Inference
• The 4 assumptions for unbiasedness, plus
homoskedasticity assumption are known as the
Gauss-Markov assumptions.
• If the Gauss-Markov assumptions hold, OLS is
BLUE.
• In order to do classical hypothesis testing, we
need to add another assumption (beyond the
Gauss-Markov assumptions).
• Assume that is independent of x1, x2,…, xk and
is normally distributed with zero mean and
variance s2: ~ iid N(0,s2)
Sisir Sarma 18.318: Introduction to Econometrics
CLM Assumptions (cont.)
• Under CLM, OLS is not only BLUE, but is
the minimum variance unbiased estimator.
• We can summarize the population
assumptions of CLM as follows
• y|x ~ Normal(b0 + b1x1 +…+ bkxk, s2)
• While for now we just assume normality,
clear that sometimes not the case.
• Large samples will let us drop normality.
Sisir Sarma 18.318: Introduction to Econometrics
The homoskedastic normal distribution with
a single explanatory variable
y
f(y|x)
. E(y|x) = b + b x
0 1
.
Normal
distributions
x1 x2
Sisir Sarma 18.318: Introduction to Econometrics
Normal Sampling Distributions
Under the CLM assumptions, conditional on
the sample values of the independent variables
ˆ
j
ˆ
b ~ Normal b ,Var b , so thatj j
bˆ bj ~ Normal0,1
j
ˆ
sd b j
ˆ
b j is distributed normally becauseit
is a linear combination of the errors
Sisir Sarma 18.318: Introduction to Econometrics
The t Test
Under the CLM assumptions
ˆ
bj b j
ˆ
se b
~ t n k 1
j
Note this is a t distribution (vs normal)
because we have to estimate s by sˆ2 2
Note the degrees of freedom : n k 1
Sisir Sarma 18.318: Introduction to Econometrics
The t Test (cont.)
• Knowing the sampling distribution for the
standardized estimator allows us to carry out
hypothesis tests
• Start with a null hypothesis
• For example, H0: bj = 0
• If accept null, then accept that xj has no effect
on y, controlling for other x’s.
Sisir Sarma 18.318: Introduction to Econometrics
The t Test (cont.)
e
To perform our test w first need to form
ˆ
bj
ˆ
" the" t statistic for b j : t bˆ
j ˆ
se b
j
We will then use our t statistic along with
o
a rejection rule to determine whether t
accept thenull hypothesis, H 0
Sisir Sarma 18.318: Introduction to Econometrics
t Test: One-Sided Alternatives
• Besides our null, H0, we need an alternative
hypothesis, HA, and a significance level
• HA may be one-sided, or two-sided
• HA: bj > 0 and HA: bj < 0 are one-sided
• HA: bj 0 is a two-sided alternative.
• If we want to have only a 5% probability of
rejecting H0 if it is really true, then we say
our significance level is 5%.
Sisir Sarma 18.318: Introduction to Econometrics
One-Sided Alternatives (cont.)
• Having picked a significance level, a, we
look up the (1 – a)th percentile in a t
distribution with n – k – 1 df and call this c,
the critical value.
• We can reject the null hypothesis if the t
statistic is greater than the critical value.
• If the t statistic is less than the critical value
then we fail to reject the null.
Sisir Sarma 18.318: Introduction to Econometrics
One-Sided Alternatives (cont.)
yi = b0 + b1xi1 + … + bkxik + i
H0: bj = 0 HA: bj > 0
Fail to reject
reject
1 a a
Sisir Sarma
0 c
18.318: Introduction to Econometrics
One-sided vs Two-sided
• Because the t distribution is symmetric, testing
H1: bj < 0 is straightforward. The critical value is
just the negative of before
• We can reject the null if the t statistic < –c, and if
the t statistic > than –c then we fail to reject the
null
• For a two-sided test, we set the critical value
based on a/2 and reject H1: bj 0 if the absolute
value of the t statistic > c
Sisir Sarma 18.318: Introduction to Econometrics
Two-Sided Alternatives
yi = b0 + b1Xi1 + … + bkXik + i
H0: bj = 0 HA: bj 0
fail to reject
reject reject
a/2 1 a a/2
Sisir Sarma
-c 0 c
18.318: Introduction to Econometrics
Summary for H0: bj = 0
• Unless otherwise stated, the alternative is
assumed to be two-sided
• If we reject the null, we typically say “xj is
statistically significant at the a % level”
• If we fail to reject the null, we typically say
“xj is statistically insignificant at the a %
level”
Sisir Sarma 18.318: Introduction to Econometrics
Testing other hypotheses
• A more general form of the t statistic
recognizes that we may want to test
something like H0: bj = aj
• In this case, the appropriate t statistic is
bˆ aj
t
j
ˆ , where
se b j
a j 0 for the standard test
Sisir Sarma 18.318: Introduction to Econometrics
Confidence Intervals
• Another way to use classical statistical testing is
to construct a confidence interval using the same
critical value as was used for a two-sided test
• A (1 - a) % confidence interval is defined as
ˆ c se b , wherec is the 1 - a percentile
bj ˆ
j
2
in a tn k 1 distribution
Sisir Sarma 18.318: Introduction to Econometrics
Computing p-values for t tests
• An alternative to the classical approach is
to ask, “what is the smallest significance
level at which the null would be rejected?”
• So, compute the t statistic, and then look up
what percentile it is in the appropriate t
distribution – this is the p-value
• p-value is the probability we would observe
the t statistic we did, if the null were true
Sisir Sarma 18.318: Introduction to Econometrics
Testing a Linear Combination
• Suppose instead of testing whether b1 is equal to a
constant, you want to test if it is equal to another
parameter, that is H0 : b1 = b2
• Use same basic procedure for forming a t statistic
ˆ ˆ
b1 b 2
t
ˆ b
se b1 ˆ
2
Sisir Sarma 18.318: Introduction to Econometrics
Testing Linear Comb. (cont.)
Since
ˆ ˆ
ˆ ˆ
se b1 b 2 Var b1 b 2 , then
Varb b Varb Varb 2Covb , b
ˆ ˆ
1
ˆ
2
ˆ
1
ˆ ˆ
2 1 2
se b b se b se b 2 s
ˆ
1
ˆ ˆ ˆ 2 2 2
1 2 1 2 12
where s is an estimate of Covb , b
12
ˆ ˆ
1 2
Sisir Sarma 18.318: Introduction to Econometrics
Testing a Linear Comb. (cont.)
• So, to use formula, need s12, which standard
output does not have
• Many packages will have an option to get it, or
will just perform the test for you
• In Eviews, after ls y c x1 x2 … xk, in the window
with the regression results, select View then
Coefficient Tests, then Wald tests to do a Wald
test of the hypothesis that b1 = b2 (type c(2) = c(3)
Sisir Sarma 18.318: Introduction to Econometrics
Example:
• Suppose you are interested in the effect of
campaign expenditures on outcomes
• Model is voteA = b0 + b1log(expendA) +
b2log(expendB) + b3prtystrA +
• H0: b1 = - b2, or H0: q1 = b1 + b2 = 0
• b1 = q1 – b2, so substitute in and rearrange
voteA = b0 + q1log(expendA) +
b2log(expendB - expendA) + b3prtystrA +
Sisir Sarma 18.318: Introduction to Econometrics
Example (cont.)
• This is the same model as originally, but
now you get a standard error for b1 – b2 = q1
directly from the basic regression
• Any linear combination of parameters
could be tested in a similar manner
• Other examples of hypotheses about a
single linear combination of parameters: b1
= 1 + b2 ; b1 = 5b2 ; b1 = -1/2b2 ; etc
Sisir Sarma 18.318: Introduction to Econometrics
Multiple Linear Restrictions
• Everything we’ve done so far has involved
testing a single linear restriction, (e.g. b1 = 0
or b1 = b2 )
• However, we may want to jointly test
multiple hypotheses about our parameters
• A typical example is testing “exclusion
restrictions” – we want to know if a group
of parameters are all equal to zero
Sisir Sarma 18.318: Introduction to Econometrics
Testing Exclusion Restrictions
• Now the null hypothesis might be
something like H0: bk-r+1 = 0, ... , bk = 0
• The alternative is just HA: H0 is not true
• Can’t just check each t statistic separately,
because we want to know if the r
parameters are jointly significant at a given
level – it is possible for none to be
individually significant at that level
Sisir Sarma 18.318: Introduction to Econometrics
Exclusion Restrictions (cont.)
• To do the test we need to estimate the “restricted
model” without xk-r+1,, …, xk included, as well as
the “unrestricted model” with all x’s included
• Intuitively, we want to know if the change in SSR
is big enough to warrant inclusion of xk-r+1,, …, xk
SSRUR r
SSRR
F , where
SSRUR n k 1
the subscript R refers to restricted and
the subscript UR refers to unrestrict ed
Sisir Sarma 18.318: Introduction to Econometrics
The F statistic
• The F statistic is always positive, since the
SSR from the restricted model can’t be less
than the SSR from the unrestricted
• Essentially the F statistic is measuring the
relative increase in SSR when moving from
the unrestricted to restricted model
• r = number of restrictions, or dfR – dfUR
• n – k – 1 = dfUR
Sisir Sarma 18.318: Introduction to Econometrics
The F statistic (cont.)
• To decide if the increase in SSR when we
move to a restricted model is “big enough”
to reject the exclusions, we need to know
about the sampling distribution of our F stat
• Not surprisingly, F ~ Fr,n-k-1, where r is
referred to as the numerator degrees of
freedom and n – k – 1 as the denominator
degrees of freedom
Sisir Sarma 18.318: Introduction to Econometrics
The F statistic (cont.)
f(F)
Reject H0 at a
fail to reject significance level
if F > c
reject
1 a a
0 c F
Sisir Sarma 18.318: Introduction to Econometrics
The R2 form of the F statistic
• Because the SSR’s may be large and unwieldy, an
alternative form of the formula is useful
• We use the fact that SSR = TSS(1 – R2) for any
regression, so can substitute in for SSRR and
SSRUR
F
R r
R
2 2
1 R n k 1
UR R
2
, where again
UR
R is restricted and UR is unrestrict ed
Sisir Sarma 18.318: Introduction to Econometrics
Overall Significance
• A special case of exclusion restrictions is to test
H0: b1 = b2 =…= bk = 0
• Since the R2 from a model with only an intercept
will be zero, the F statistic is simply
2
R k
F
1 R n k 1
2
Sisir Sarma 18.318: Introduction to Econometrics
General Linear Restrictions
• The basic form of the F statistic will work
for any set of linear restrictions
• First estimate the unrestricted model and
then estimate the restricted model
• In each case, make note of the SSR
• Imposing the restrictions can be tricky –
will likely have to redefine variables again
Sisir Sarma 18.318: Introduction to Econometrics
F Statistic Summary
• Just as with t statistics, p-values can be
calculated by looking up the percentile in
the appropriate F distribution
• If only one exclusion is being tested, then F
= t2, and the p-values will be the same
• You can use Wald test to test multivariate
hypotheses as well
Sisir Sarma 18.318: Introduction to Econometrics
