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计量经济学Lecture 6-1


Heteroskedasticity
Heteroskedasticity Robust Standard Error BP test White test Weighted Least Squares (WLS) Estimators Generalized Least Squares (GLS) Estimators

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Route of Today

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Release one of our CLM assumptions ------Homoskedasticity What happen if so? How do we test the hetroskedasiticity? How to solve Hetroskedasiticity problem?

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Assumption discusses:
CLM Assumptions:

MLR.1:Linear in parameters MLR.2:Random Sampling(Exogenously generated data) MLR.3:Zero Conditional Mean(Exogeneity of the independent variables) MLR.4:No perfect collinearity(Fulll Rank of X) MLR.5:Homoskedasticiy and nonautocorrelation MLR.6:Normal distribution: the error term are normally distributed
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Heteroskedasticity
y = b0 + b1x1 + b2x2 + . . . bkxk + u

Var (ui | X i ) ? ? i ? cons tan t
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What is Heteroskedasticity?
If homoskedasiticity is not true, that is if the variance of u is different for different values of the x’s, then the errors are heteroskedastic ? Example: 1.Estimating returns to education and ability is unobservable, and think the variance in ability differs by educational attainment 2.Estimating saving on income.
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Example of Heteroskedasticity
f(y|x)

.
x1 x2 x3

.

.

E(y|x) = b0 + b1x

x
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Variance with Heteroskedasticity
For the simple regression case, ? ? b ? ? ? xi ? x ?ui , b1 1 2 ? ?xi ? x ?

? so Var b1 ?

? ?

? ?x

? x ? ? i2 i , 2 SSTx
2 2

where SSTx ? ? ? xi ? x ?

A valid estimator for this when ? i2 ? ? 2is

? ?x

? ? x ? ui2 i ? , where ui are the OLS residuals 2 SSTx
2

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Variance with Heteroskedasticity
For the general multiple regression model, ? a valid estimator of Var b with heterosked asticity is

? ? ? ? r? u ? ? Var ?b ? ? SSR
j j

2 2 ij i 2 j

,

?ij is the i th residual from where r regressing x j on all other independen t variables, and SSRj is the sum of squared residuals from this regression
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If there exists Heteroscedasticity,
1. OLS estimator is not the smallest variance. E.g.
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Yi ? b 0 ? b1 X i ? ui if Var (ui ) ? ? X
2 2

2 i

?)?? Var ( b
1

?x X (? x )

2 2 i i 2 2 i

?x ? (? x )
?
2

2 i 2 2 i

Heteroscedasticity

Homoscedasticity
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If there exists Heteroskedasticity,
? ?

2. Hypothesis test error E.g H : b ? 0
0 1

t ?

? b1 ? Var ( b1 )

overvalued

3. Difficult to get confidence interval

? ? b1 ? b1 ? ta (n ? k ) Var ( b1 )
2

4. Confidence interval of explained variable is predicted uncorrectly
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Why Worry About Heteroskedasticity?
OLS is still unbiased and consistent, even if we do not assume homoskedasticity ? The standard errors of the estimates are biased if we have heteroskedasticity ? If the standard errors are biased, we can not use the usual t statistics or F statistics for testing our hypothesis.
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To find the Heteroskedasticity.
1. Residual graph E.g. SAVING.DTA 2. Goldfeld-Quandt test

n?c ? e /( 2 ? k ) ? e22i F? ? 2 n?c 2 ? e1i ? e1i /( 2 ? k )
2 2i
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3. The Breusch-Pagan Test
Essentially want to test H0: Var(u|x1, x2,…, xk) = s2, which is equivalent to H0: E(u2|x1, x2,…, xk) = E(u2) = s2 ? If assume the relationship between u2 and xj will be linear, can test as a linear restriction ? So, for u2 = d0 + d1x1 +…+ dk xk + v this means testing H0: d1 = d2 = … = dk = 0
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The Breusch-Pagan Test (cont.)
Don’t observe the error, but can estimate it with the residuals from the OLS regression ? After regressing the residuals squared on all of the x’s, can use the R2 to form an F or LM test ? The F statistic is just the reported F statistic for overall significance of the regression F = [R2/k]/[(1 – R2)/(n – k – 1)], which is distributed Fk, n – k - 1 ? The LM statistic is LM = nR2, which is distributed c2k
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? Eg. HPRICE1.dTA
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Steps of B-P test

Yt ? b 0 ? b1 X 1t ? ...... ? b k X kt ? ui Var (ut ) ? ? ? ? 0 ? ?1Z1 ? ...... ? ? p Z p
2 t

Z? X

H 0 : ?1 ? ? 2 ? ...... ? ? p ? 0 H1 : ?1 ? 0,? 2 ? 0......,? p ? 0
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Steps of B-P test (cont.)

? ? ? et ? Yt ? b 0 ? b1 X 1t ? ...... ? b k X kt

?e ? ? ?
2

2 t

n

e ? ? 0 ? ?1Z1 ? ...... ? ? p Z p ? Vt ?2 ? ESS 2 ~ ? ( p) 2
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2 t

4.The White Test
The Breusch-Pagan test will detect any linear forms of heteroskedasticity ? The White test allows for nonlinearities by using squares and crossproducts of all the x’s ? Still just using an F or LM to test whether all the xj, xj2, and xjxh are jointly significant
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Alternate form of the White test
Consider that the fitted values from OLS, ?, are a function of all the x’s ? Thus, ?2 will be a function of the squares and crossproducts and ? and ?2 can proxy for all of the xj, xj2, and xjxh, so ? Regress the residuals squared on ? and ?2 and use the R2 to form an F or LM statistic ? Note only testing for 2 restrictions now
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Robust Standard Errors
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We have a consistent estimate of the variance, the square root can be used as a standard error n for inference rij2ui2 ?? ? ? ? Var ( b j ) ? i ?1 2 SSRj Typically call these robust standard errors, sometimes called White,Huber,or Eicker standard error. Sometimes the estimated variance is corrected for degrees of freedom by multiplying by n/(n – k – 1) 19 As n → ∞ it’s all the same, though

Robust Standard Errors (cont)
Important to remember that these robust standard errors only have asymptotic justification – with small sample sizes t statistics formed with robust standard errors will not have a distribution close to the t, and inferences will not be correct ? In Stata, robust standard errors are easily obtained using the robust option of reg as: reg income age edu exp….., robust
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A Robust LM Statistic
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Run OLS on the restricted model and save the residuals ? Regress each of the excluded variables on all of the included variables (q different regressions) and save each set of residuals ?1, ?2, …, ?q Regress a variable defined to be = 1 on ?1 ?, ?2 ?, …, ?q ?, with no intercept The LM statistic is n – SSR1, where SSR1 is the sum of squared residuals from this final regression
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Weighted Least Squares
While it’s always possible to estimate robust standard errors for OLS estimates, if we know something about the specific form of the heteroskedasticity, we can obtain more efficient estimates than OLS ? The basic idea is going to be to transform the model into one that has homoskedastic errors – called weighted least squares
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If the Heteroskedasticity is Known
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Let x denote all the explanatory variables and assume:

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Var(u | x) ? ? h( x) Where h(x ) is some function of the explanatory variables that determines the heteroskedasticity. And it is needed: h( x) ? 0
2

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Example
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The simple saving function:

savi ? b 0 ? b1inci ? ui Var(ui | inci ) ? ? inci
2

here : h( x ) ? h(inc) ? inc
?

Means the variance of the error is proportional to the level of income.

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Then
yi ? b 0 ? b1 xi1 ? b 2 xi 2 ? ...... ? b k xik ? ui Var(ui | xi ) ? E (u | xi ) ? ? hi
2 i 2

Var(ui yi or

hi ) ? E ((ui

hi ) 2 ) ? ? 2 hi hi ? ? 2 hi ) ? b 2 ( xi 2 hi ) ? ...... ? b k ( xik hi ) ? ui hi

Now take the weight : hi ? b 0 hi ? b1 ( xi1

y *i ? b 0 x*i 0 ? b1 x*i1 ? b 2 x*i 2 ? ...... ? b k x*ik ? u *i
These are examples of generalized least squares(GLS) estimators. In this case, we call weighted least square(WLS) estimators
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Weighted Least Squares(Cont.)
eg. Wi ? 1 let X ?
* * i

?

2 i

, i ? 1,2.....n
i i

?W X ?W
i * i

Y

*

?

?W Y ?W
i i *

i

x ? Xi ? X y ? Yi ? Yi ? ? then b * ? Y * ? b X *
* i 0

Wi yi* xi* ? b i* ? ? Wi ( xi* ) 2 ?
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Generalized Least Squares
Estimating the transformed equation by OLS is an example of generalized least squares (GLS) ? GLS will be BLUE in this case ? GLS is a weighted least squares (WLS) procedure where each squared residual is weighted by the inverse of Var(ui|xi)
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Weighted Least Squares
While it is intuitive to see why performing OLS on a transformed equation is appropriate, it can be tedious to do the transformation ? Weighted least squares is a way of getting the same thing, without the transformation ? Idea is to minimize the weighted sum of squares (weighted by 1/hi)
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More on WLS

WLS is great if we know what Var(ui|xi) looks like. ? The problem is: in most cases, we don’t know the form of heteroskedasticity.
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Case of form being known up to a multiplicative constant
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Suppose the heteroskedasticity can be modeled as Var(u|x) = s2h(x), where the trick is to figure out what h(x) ≡ hi looks like E(ui/√hi|x) = 0, because hi is only a function of x, and Var(ui/√hi|x) = s2, because we know Var(u|x) = s 2 hi So, if we divided our whole equation by √hi we would have a model where the error is homoskedastic
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Feasible GLS
More typical is the case where you don’t know the form of the heteroskedasticity ? In this case, you need to estimate h(xi) ? Typically, we start with the assumption of a fairly flexible model, such as ? Var(u|x) = s2exp(d0 + d1x1 + …+ dkxk) ? Since we don’t know the d, must estimate
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Feasible GLS (continued)
Our assumption implies that u2 = s2exp(d0 + d1x1 + …+ dkxk)v ? Where E(v|x) = 1, then if E(v) = 1 ? ln(u2) = a0 + d1x1 + …+ dkxk + e ? Where E(e) = 1 and e is independent of x ? Now, we know that ?is an estimate of u, so we can estimate this by OLS
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Feasible GLS (continued)
Now, an estimate of h is obtained as ? = exp(?), and the inverse of this is our weight ? Run the original OLS model, save the residuals, ? square them and take the log , 2 ? Regress ln(? ) on all of the independent variables and get the fitted values, ? ? Do WLS using 1/exp(?) as the weight
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WLS Wrapup
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When doing F tests with WLS, form the weights from the unrestricted model and use those weights to do WLS on the restricted model as well as the unrestricted model Remember we are using WLS just for efficiency – OLS is still unbiased & consistent Estimates will still be different due to sampling error, but if they are very different then it’s likely that some other Gauss-Markov assumption is false
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? ?

8.4,8.6,8.7 C8.2, C8.7



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