Economic STATA homework
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homeworkhelp.pptxDistribution of Estimates
Linear Regression Model
Assume (yt, xt) are independent and identically distributed and E(xtet) = 0
Estimation Consistency
The estimates approach the true values as the sample size increases.
Estimation variance decreases as the sample size increases.
Illustration of Consistency
Take a random sample of U.S. men
Estimate a linear regression of log(wages) on education
Total sample = 9089
Start with 100 observations, and sequentially increase sample size until in the final regression use the whole 9089.
Sequence of Slope Coefficients
Asymptotic Normality
4
Illustration of Asymptotic Normality
Time Series
Do these results apply to time-series data?
Consistency
Asymptotic Normality
Variance Formula
Time-series models
AR models, i.e., xt = yt-1
Trend and seasonal models
One-step and multi-step forecasting
Derivation of Variance Formula
For simplicity
Assume the variables have zero mean
The regression has no intercept
Model with no intercept:
Model with no intercept
OLS minimizes the sum of squares
The first-order condition is
Solution
Now substitute
We have
The denominator is the sample variance (when x has mean zero), so
10
Then
Where
Since
Then
From the covariance formula
When the observations are independent, the covariances are zero.
And since
We obtain
We have found
As stated at the beginning.
Extension to Time-Series
The only place in this argument where we used the assumption of the independence of observations was to show that vt = xtet has zero covariance with vj = xjej.
This is saying that vt is not autocorrelated.
Unforecastable one-step errors
In one-step-ahead forecasting, if the regression error is unforecastable, then vt is not autocorrelated.
In this case, the variance formula for the least-squares estimate is
Why is this true?
The error is unforecastable if
For simplicity, suppose that xt = 1.
Then for
Summary
In one-step-ahead time-series models, if the error is unforecastable, then least-squares estimates satisfy the asymptotic (approximate) distribution
As the sample size T is in the denominator, the variance decreases as the sample size increases.
This means that least-squares is consistent.
Variance Formula
The variance formula for the least-squares estimate takes the form
This formula is valid in time-series regression when the error is unforecastable.
Classical Variance Formula
If we make the simplifying assumption
Then
Homoskedasticity
The variance simplification is valid under “conditional homoskedasticity”
This is a simplifying assumption made to make calculations easier, and is a conventional assumption in introductory econometrics courses.
It is not used in serious econometrics.
Variance Formula: AR(1) Model
Take the AR(1) model with unforecastable homoscedastic errors
Then the variance of the OLS estimate is
Since in this model
AR(1) Asymptotic Variance
We know that
So
The asymptotic distribution is very simple
The variance is a function of the unknown true value of
As || increases, the variance decreases, so the OLS estimate is actually more precise.
Distribution of Least Squares
In classic regression, if the errors are iid normal, and independent of the regressors, then the least-squares estimates have an exact normal distribution, not just asymptotic.
This is not true in most time-series regressions.
Non Classical Distributions
Estimates in autoregressive models
Biased downwards
Skewed
Thick tails
Especially
When autoregressive coefficients are large
Sample sizes are small
These issues diminish in large samples
Interpretation
Estimates of autoregressive parameters are random.
Even if the regression error is normal, the parameter estimates are not normally distributed.
Distributions are less normal when AR coefficient is large.
Distributions are more concentrated and normal when sample size is large.
Asymptotic Standard Deviation
The least-squares estimate is asymptotically (approximately) normally distributed.
In the simple model
Then
The standard deviation measures the precision of the estimate, but it is unknown.
Standard Errors
Estimates of the standard deviations are called standard errors, and are reported in the regression output.
They are used to measure precision.
Classical standard errors
A classical standard error is an estimate of the standard deviation from the formula
This formula is valid under conditional homoskedasticity
This last equation is unforecastability of the variance. This is a particularly poor assumption for financial data.
Robust Standard Errors
“Robust” standard errors are estimates of
These are conventional standard errors for regression analysis
Due Halbert White (1980). Most referenced paper in economics.
Robust standard errors will often differ by quite a lot from estimates of standard errors that use the assumption of homoskedasticity.
Computation
In STATA, the default is homoskedastic standard errors.
They are reported automatically with the regress command.
For robust standard errors, use the “r” option:
.reg rgdp L.rgdp, r
Example: Real GDP Growth (classical)
Real GDP Growth (robust)
Issue
With the “r” option, STATA does not report the sum of squared error table. You might want to see this, so you might want to run both command:
.reg y x
.reg y x, r
Interpretation of standard errors
The standard errors measure the precision of the estimate.
Small standard errors mean the estimate is precise, which is good for forecasting.
Large standard errors mean the estimate is not precise, which can lead to inaccurate forecasts.
Interpretation of t-statistics
“t” is the coefficient estimate divided by the standard error.
It is used to test if the coefficient is zero.
“P”>t is the p-value of the t-statistic
If p<.05, you reject the hypothesis of a zero coefficient
Hypothesis tests are useful for assessing economic theories, but are less useful for picking good forecast models.
The 95% confidence interval is the coefficient estimate plus and minus 1.96 times the standard error. Helps to gauge possible values of the true coefficient.
Summary
In one-step-ahead forecast regressions with unforecastable errors, robust standard errors are generally appropriate. Classical standard errors are appropriate under conditional homoskedasticity.
Next class October 16
Complete reading from Wooldridge. Topic is autocorrelation and heteroskedastic consistent standard errors.
Sequence of Slope Coefficients
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0 2000 4000 6000 8000 10000 observation
Sequence of Slope Coefficients.
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observation
Illustration of Asymptotic Normality
Illustration of Asymptotic Normality
_cons 2.154868 .3415603 6.31 0.000 1.482516 2.827221 L4. -.0719696 .0592348 -1.21 0.225 -.1885718 .0446325 L3. -.0893879 .0621447 -1.44 0.151 -.211718 .0329422 L2. .1695538 .0622018 2.73 0.007 .0471113 .2919963 L1. .3204166 .0595748 5.38 0.000 .2031452 .437688 rgdp
rgdp Coef. Std. Err. t P>|t| [95% Conf. Interval]
Total 4207.796 284 14.8161831 Root MSE = 3.5565 Adj R-squared = 0.1463
Residual 3541.53535 280 12.6483405 R-squared = 0.1583 Model 666.260654 4 166.565164 Prob > F = 0.0000
F(4, 280) = 13.17 Source SS df MS Number of obs = 285
. regress rgdp L(1/4).rgdp
.
_cons 2.154868 .4135065 5.21 0.000 1.340892 2.968845 L4. -.0719696 .0735299 -0.98 0.329 -.2167112 .0727719 L3. -.0893879 .0694964 -1.29 0.199 -.2261897 .0474138 L2. .1695538 .0819025 2.07 0.039 .0083309 .3307766 L1. .3204166 .0733727 4.37 0.000 .1759846 .4648487 rgdp
rgdp Coef. Std. Err. t P>|t| [95% Conf. Interval] Robust
Root MSE = 3.5565 R-squared = 0.1583 Prob > F = 0.0000 F(4, 280) = 9.83 Linear regression Number of obs = 285
. regress rgdp L(1/4).rgdp, r
