Inference and Estimation: An Econometrics Primer

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Inference and Estimation An Econometrics Primer

• Bryan Lohmar
• Economic Research Service

2
What is Econometrics?

• Measuring relationships with observational data
  (my definition)
• Econ Economics
• Metrics Measurement
• Why the emphasis on observational data?

3
History of Econometrics

• Regression analysis developed for experimental
  data
• Grilliches uses this technique to estimate
  fertilizer response on observational data
• Hidden relationships become an issue

4
Why Economics?

• All social sciences use observational data
• Economics provides a powerful tool for positing
  relationships, thus is critical to good
  Econometrics

5
Econometrics is all about the Mean

• The mean of a random sample size n is
• And its variance is

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2 Important Properties of the Sample Mean

• It is and unbiased estimate for the underlying
  population mean
• And it is the minimum variance unbiased estimator
• The equal weights (1/n) given to each observation
  in the sample yields the estimator with the
  smallest variance

7
Econometrics Begins When We Go One Step Further

• Model the mean of Y as a function of other
  variables
• Eyixi ß0 ß1xi
• Or
• yi ß0 ß1xi ui

8
OLS Parameter Estimates

• The OLS estimate of the parameter, ß1, is
• This estimate is unbiased so long as.

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Requirements of Unbiasedness

• Correct linear model specification
• Sample is randomly drawn
• Euixi 0, zero conditional mean, no
  correlation between the us and the xs.
• 4) There exists variation in the xs the
  denominator term is not zero

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Variance of the OLS Estimates

• The variance of the OLS parameter estimate is
• Where s2 is the variance of the error term of the
  model gt Varui
• This is the smallest variance of all the unbiased
  parameter estimators so long as there is
  homoskedasticity

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These Principles Extend to the Multivariable
Models

• If the model has more than on independent
  variable
• Eyix1i,,xki ß0 ß1x1i ßkxki
• Or
• yi ß0 ß1x1i ßkxki ui
• Then..

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The OLS Parameter Estimates Change Slightly

• The OLS estimate for the parameter ßj, associated
  with the independent variable xj, is
• Where,

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The Variance of the Parameter, However, Can
Change Significantly

• The variance of the OLS estimate for the
  parameter ßj, associated with the independent
  variable xj, is
• Where Rj2 is from the regression of xj on all the
  other independent variables

14
Omitted Variable Bias The Biggest Problem in
Econometrics

• Simple example, you estimate the model
• But the true model is
• The estimate is biased, and the bias is
  equivalent to where the delta comes from

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Fundamental Trade-OffBias Versus Efficiency

• Suppose you estimate the models
• and
• Then, to the extent that x1 and x2 are
  correlated,
• Var lt Var
• Moreover, if ? 0, then is biased, but if
  0, then both estimates are unbiased and
  has a smaller variance.

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STATA Output wages, education experience and
tenure
. desc wage educ exper tenure
storage display value variable name type
format label variable
label --------------------------------------------
----------------------------------- wage
float 8.2g average hourly
earnings educ byte 8.0g
years of education exper byte
8.0g years potential
experience tenure byte 8.0g
years with current employer . sum wage
educ exper tenure Variable Obs
Mean Std. Dev. Min
Max ---------------------------------------------
------------------------ wage 526
5.896103 3.693086 .53 24.98
educ 526 12.56274 2.769022
0 18 exper 526
17.01711 13.57216 1 51
tenure 526 5.104563 7.224462
0 44
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Model 1 Wagef(educ)
reg wage educ Source SS df
MS Number of obs
526 -------------------------------------------
F( 1, 524) 103.36 Model
1179.73204 1 1179.73204 Prob gt F
0.0000 Residual 5980.68225 524
11.4135158 R-squared
0.1648 ------------------------------------------
- Adj R-squared 0.1632 Total
7160.41429 525 13.6388844 Root
MSE 3.3784 ------------------------------
------------------------------------------------
wage Coef. Std. Err. t
Pgtt 95 Conf. Interval ------------------
--------------------------------------------------
--------- educ .5413593 .053248
10.17 0.000 .4367534 .6459651
_cons -.9048516 .6849678 -1.32 0.187
-2.250472 .4407687 ----------------------------
--------------------------------------------------

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Model 2 Wagef(educ, exper)
reg wage educ exper Source SS
df MS Number of obs
526 -------------------------------------------
F( 2, 523) 75.99 Model
1612.2545 2 806.127251 Prob gt F
0.0000 Residual 5548.15979 523
10.6083361 R-squared
0.2252 ------------------------------------------
- Adj R-squared 0.2222 Total
7160.41429 525 13.6388844 Root
MSE 3.257 ------------------------------
------------------------------------------------
wage Coef. Std. Err. t
Pgtt 95 Conf. Interval ------------------
--------------------------------------------------
--------- educ .6442721 .0538061
11.97 0.000 .5385695 .7499747
exper .0700954 .0109776 6.39 0.000
.0485297 .0916611 _cons -3.390539
.7665661 -4.42 0.000 -4.896466
-1.884613 ----------------------------------------
--------------------------------------
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Relationship between education and wage after
controlling for experience
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Experience and Education
. reg exper educ Source SS
df MS Number of obs
526 -------------------------------------------
F( 1, 524) 51.65 Model
8677.05239 1 8677.05239 Prob gt F
0.0000 Residual 88029.7936 524
167.995789 R-squared
0.0897 ------------------------------------------
- Adj R-squared 0.0880 Total
96706.846 525 184.203516 Root
MSE 12.961 ------------------------------
------------------------------------------------
exper Coef. Std. Err. t
Pgtt 95 Conf. Interval ------------------
--------------------------------------------------
--------- educ -1.468182 .2042881
-7.19 0.000 -1.869507 -1.066858
_cons 35.4615 2.627905 13.49 0.000
30.29898 40.62402 ----------------------------
--------------------------------------------------
The negative relationship between education and
experience results in a downward bias on the
coefficient for education when experience is
omitted (model 1), then the coefficient on
education increases in model 2 when experience is
included.
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Model 3 Wagef(educ, exper, tenure)
. reg wage educ exper tenure Source
SS df MS Number of obs
526 ---------------------------------------
---- F( 3, 522) 76.87
Model 2194.1116 3 731.370532
Prob gt F 0.0000 Residual
4966.30269 522 9.51398984 R-squared
0.3064 ------------------------------------
------- Adj R-squared 0.3024
Total 7160.41429 525 13.6388844
Root MSE 3.0845 -------------------------
--------------------------------------------------
--- wage Coef. Std. Err. t
Pgtt 95 Conf. Interval ----------------
--------------------------------------------------
----------- educ .5989651 .0512835
11.68 0.000 .4982176 .6997126
exper .0223395 .0120568 1.85 0.064
-.0013464 .0460254 tenure .1692687
.0216446 7.82 0.000 .1267474
.2117899 _cons -2.872735 .7289643
-3.94 0.000 -4.304799 -1.440671 -----------
--------------------------------------------------
----------------- Adding tenure, which is
positively correlated with experience, changes
the coefficient on experience and its standard
error. Since tenure is positively correlated
with wage and experience, the bias on the model
that omits tenure is positive, so the coefficient
decreases when tenure is added. However, tenure
and experience often have identical types of
effects, so if you want to estimate the
relationship between experience and wage,
including tenure might not be a good idea.
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Experience and Tenure
. reg tenure exper Source SS
df MS Number of obs
526 -------------------------------------------
F( 1, 524) 174.01 Model
6830.91075 1 6830.91075 Prob gt F
0.0000 Residual 20570.3383 524
39.2563708 R-squared
0.2493 ------------------------------------------
- Adj R-squared 0.2479 Total
27401.249 525 52.1928553 Root
MSE 6.2655 ------------------------------
------------------------------------------------
tenure Coef. Std. Err. t
Pgtt 95 Conf. Interval ------------------
--------------------------------------------------
--------- exper .2657729 .0201477
13.19 0.000 .2261926 .3053532
_cons .581876 .4383861 1.33 0.185
-.2793342 1.443086 ----------------------------
--------------------------------------------------

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Inference

• Lastly, what do we mean by inference?
• What is Minnesota like?
• You can infer this from a sample of photographs
• You cannot prove anything, but you can infer that
  a specific hypothesis either may be true (cannot
  reject hypothesis) or not likely true (reject
  hypothesis).
• Hypothesis Minnesota is a dry, desert state.
• Reject hypothesis based on a random sample of
  photographs