Yt = + Xt + t
Ordinary Least Square (OLS) Estimators
• It is given that the economic theory assumes a linear model between the dependent and independent explanatory variable, with some random (unexplained) deviations/errors/residuals:
Ct = ( + ( Yt + (t
=> (t = (Ct - ( - ( Yt)
• We dislike errors.
• Our dissatisfaction from the errors increases very rapidly. If the error to either side doubles, our dissatisfaction increases more than double.
• Therefore, we would like to impose some restrictions on our econometric model regarding the errors:
(I) (t (t = 0 - There is no systematic (aggregate) error.
(II) Min (t (t2 - Penalize for larger errors to either side.
• It is easy to prove that: given our theoretical linear model, imposing the above restrictions provides us with the Best Linear Unbiased Estimators. In other words, the OLS estimators are BLUE (Gauss-Markov theorem).
• What can we infer from these restrictions? How can we use them in order to derive the estimators- the equations that estimate our coefficients: ( & (? *
• Before we proceed, notice:
1) Our sample includes n observation, and our regression runs m independent variables.
2) When we drop the subscript t this means we are referring to the mean of the variable
3) When we use lower-case letters, then we refer to the deviation from the mean of that letter.
4) We want to minimize the vertical deviations from the regression line. That is different from minimizing the horizontal deviations from the regression line.
5) The regression provides us with estimates for the correlation- its sign, magnitude and significance. However, further economics and econometric theory is need for identifying the causality.
(I) (t (t = 0
(a) (t (Ct - ( - ( Yt) = 0
C - ( - ( Y = 0
C = ( + ( Y
( = C - ( Y - The constant coefficients
(b) (t (t / n = 0
( = 0 .
(II) Min (t (t2
(a) (t = Ct - ( - ( Yt - By the model specification (see page 1)
0 = C - ( - ( Y - Our 1st conclusion from 1st restriction (see page 2)
=> (t = (Ct – C) - ( (Yt – Y) ( ct - ( yt
(t = ct - ( yt .
(b) Min (t (t2 = Min (t (ct - ( yt) 2
= Min (t (ct 2 - 2( ct yt + ( 2 yt 2)
• ( minimizes the Sum of Squared Errors (SSE) should be such that the first derivative equals to zero (FOC):
(t (0 - 2 ct yt + 2 ( yt 2) = 0
(t 0 - 2 (t ct yt + 2 ( (t yt 2 = 0
( = (t ct yt / (t yt 2 ( ( = Scy / Syy .
( = [(t ct yt / n] / [(t yt 2 /n] ( ( = (cy2 / (y2 .
( = [(cy2 / (y (c] [(c / (y] ( ( = ( cy2 * [(c / (y] .
When ( cy , ( 0 ( R2 ( 1
Summary:
• The theoretical economic model assumes linear relation in levels:
Ct = ( + ( Yt + (t => (t = (Ct - ( - ( Yt)
• The OLS econometric model imposes the following:
(I) (t (t = 0
(II) Min (t (t2
• Gauss-Markov theorem: OLS estimators are BLUE.
• Estimators:
( = (t ct yt / (t yt 2 ( ( = Scy / Syy .
( = [(t ct yt / n] / [(t yt 2 /n] ( ( = (cy2 / (y2 .
( = [(cy2 / (y (c] [(c / (y] ( ( = ( cy2 * [(c / (y] .
( = C - ( Y
• (1-()% Confidence interval: (= b ( Sb t(n-m -1,1-(/2 ) ,where Sb =( [(t et2/(t yt2]. If t-statistic(|b/Sb| < t(n-m-1,1-(/2 ) then there is NO statistically significant linear relation between Ct and Yt.
• Linear Fitness: R2 = 1 - (t (t2 / (t ct2, 0 ( R2 ( 1
• A numerical example:
|Observation # |Ct |Yt |ct |yt |ct2 |yt2 |ct*yt |Cte |e |e2 |
| | | | | | | | | | | |
|1 |100 |137 | | 20|9 |400 | 60|112 |-12 |149 |
| | | |3 | | | | | | | |
|2 |90 |115 | | |49 |4 | 14|95 |-5 |30 |
| | | |(7) |(2) | | | | | | |
|3 |75 |92 | (22)| (25) |484 |625 | 550 |78 |-3 |9 |
|4 |110 |120 | 13| |169 |9 | 39|99 |11 |115 |
| | | | |3 | | | | | | |
|5 |104 |116 | 7 | |49 |1 | |96 |8 |60 |
| | | | |(1) | | |(7) | | | |
|6 |120 |142 | 23| 25|529 |625 | 575 |116 |4 |16 |
|7 | 80 |97 | (17)| (20)|289 |400 | 340 |82 |-2 |3 |
| | | | | | | | | | | |
|Sum |679 |819 | | |1578 |2064 |1571 |679 |0 |382 |
|Average |97 |117 | | |225 |295 |224 |97 |0 |55 |
Therefore,
|(c2 |= |225 | |Scc |= |1578 |
|(y2 |= |295 | |Syy |= |2064 |
|(yc2 |= |224 | |Syc |= |1571 |
|(cy |= |0.93 | |Sb |= |0.43 |
|( |= |0.76 | |t-statistic = |1.77 |
|( |= |7.95 | |t(n-m-1,1-(/2 ) = |2.571 |
|R2 |= |0.76 | |95% Confident Interval? |Is it significant? |
* All the above are estimates therefore, add ^ for all.
⇨ Caveats:
• (I) Spurious Regression: Due to the trend nature of many macroeconomic time series data, one should be warrant that a spurious strong linear correlation might be found from running a linear regression, whether or not there is really any regression at work. [Can the Somalian population explain the increase in the US GDP since both have been increasing?]. Therefore, note that the independent variable should help us to estimate the deviation of the dependent variable from its drift (which is just its mean if the variable is stationary).
• (II) Missing variables: notice that if the regression does not include a variable that has significant effect on the dependant variable, and this missing variable is correlated with the included independent variable, then the effect of the missing variable will show up via the included independent variable, and therefore, it’s coefficient will be biased. Example & Proof!
• (III) Miss specification: it is important not only to pick up the right independent variables, but also the right formation of relationship. In other words, econometrist has to choose the right function. For example, he has to decide whether it is linear in levels or in logs!
Elasticity
(a) Definition
( C,Y = The percentage change of C due to one percentage change in Y
= % (C / % (Y
= [(C/C] / [(Y/Y]
( C,Y = [(C/(Y] * [Y/C]
(b) If the economic theory assumes an exponential model (instead of previous linear model), then:
Ct = A Yt( e(t
• Therefore, in order to estimate our theoretical model we can run log_log model:
ln(Ct) = Ln(A) + ( ln((Yt) + (t
where: ( = ( ln(Ct) / ( ln(Yt)
• Which means that, instead of assuming a constant propensity to consume (and increasing elasticity of consumption with respect to the disposal income) as in the linear model, the exponential model assumes a constant elasticity of consumption with respect to the disposal income (and decreasing propensity to consume).
Proof: Apply the chain rule:
( = [( ln(Ct) / (C] * [(C/(Y] * [(Y / ( ln(Yt)] = [1/C] * [(C/(Y] * [Y / 1]
= [(C/(Y] * [Y / C] = ( C,Y
t-distribution (from Green)
-----------------------
(
(
(t
Yt
Ct
Covariance(Ct ,Yt)
Sum of Squares
Correlation(Ct ,Yt) 2 (1
Covariance(Ct ,Yt)
Sum of Squares
Ct
Yt
0
[pic]
Variance(Yt) = STD(Yt)2
Variance(Yt) = STD(Yt)2
Correlation(Ct ,Yt) 2 (1
Ct
Cte
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