Question

Elaborate on the concept of consistency, homoskedasticity, and efficiency in an econometric model (regression).

Answer 1

For Regression, y = a + bx.

Under assumptions,

• The regression model is linear in parameters.
• The mean of residuals is zero.
• Homoscedasticity of residuals or equal variance.
• No autocorrelation of residuals.
• The X variables and residuals are uncorrelated.
• The variability in X values is positive.
• The regression model is correctly specified.
• No perfect multicollinearity.

Consistency: When the estimates and converge to true estimates a and b asymptotically for large sample or when n -> infinity.

So, that is as close to y as possible.

It relates with being unbiased estimator in the long run to give unbiased results.

Heteroskedasticity: Heteroscedasticity means unequal scatter. In regression analysis, we talk about heteroscedasticity in the context of the residuals or error term. Specifically, heteroscedasticity is a systematic change in the spread of the residuals over the range of measured values. Heteroscedasticity is a problem because ordinary least squares (OLS) regression assumes that all residuals are drawn from a population that has a constant variance (homoscedasticity).

To test this, we plot the residuals ei = i - yi against predicted values . The plot should exhibit random pattern and if there comes to be an identifiable pattern , that indicates heteroskedasticity.

The below image shows no pattern (or homoscedasticity).

Efficiency: efficiency is a measure of quality of an estimator, of an experimental design, or of a hypothesis testing procedure.

Here, we can compare the efficiency of parameters under Hypothesis testing and model significance using ANOVA under experimental design.

Let T denote estimate of a and b (theta) here i.e., and

Let T be an estimator for the parameter θ. The mean squared error of T is the value.

Here,

Therefore, an estimator T1 performs better than an estimator T2 if {\displaystyle MSE(T_{1})<MSE(T_{2})}

Then, for b1 and b2 two estimates of , we choose the one with minimum variance or greater efficiency.

Note : All these characteristics make the parameter estimates BLUE (Best - minimum variance, L - linear, U - unbiased , E - estimate)

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