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Two Stage Model with Endogenous Regressor (Two stage least squares)

Two Stage Model with Endogenous Regressor (Two stage least squares)

If you suspect a variable to be endogenous because it is determined within the system or that something else causes it, then you will want to (at a bare minimum) run a two stage least squares model. It’s called a TSLS model because there are two stages. The first stage regresses the suspectedendogenous variable on all independent variables and what is called an instrument. Instruments are correlated with the endogenous variable but not with the error term. Run this model, then collect the predicted values. These will be used in the second stage.

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First Stage

  • Regress the suspected endogenous variable on all the independent variables and any instruments.
  • You will want to collect the predict values from the first stage. These values will be used in the second stage of the model.
  • In the second stage, regress the dependent variable on all the original predictor variables and include the predicted values from the first stage.

 

Second Stage

  • Take the predicted values and use them in place of the endogenous variable in the main model. The coefficients you get will be consistent and unbiased estimates (assuming the model is specified appropriately) but will not be efficient since the standard errors are wrong.

Correcting Standard Errors

The standard errors produced in the second stage will be wrong, so you need to apply a correction factor to them. To do this, you will need a few pieces of information and calculated residuals, not the ones provided to you by Excel.

  • Take the original data (including the endogenous variable) and place it near the regression output.
  • Find the residuals by multiplying the coefficients by their true values using the following:

Residuals = y – a0 – a1*X1 – a2*X2 – … – an*Xn

Where y = the dependent variable

a0 = the intercept or constant

a1 = the parameter estimate of the predicted variable and X1 = the endogenous variable data

a2 = the parameter estimate of the second variable and X2 = the data for the second variable

an = the parameter estimate of the nth variable and Xn = the data for the nth variable

  • Next, square the residuals from step 2 and sum them.
  • Divide the squared residuals by the (n – # of variables in model)
  • Take the square root of step 5
    1. You can do step 3 – 5 in one step by using

=SQRT(SUMSQ(range of obs.)/(n-regression df))

  • Next, find the square root of the mean squared error (MS residual)
  • Divide the step 5 result by the step 6 result. This is the correction factor that needs to be applied to the standard errors for each variable in the second stage of the regression.
  • Copy the regression output and paste it near the correction factor you calculated.
  • Correct the standard errors by multiplying the second stage standard error by the correction factore.
  • Correct the t-stats by dividing the second stage coefficients by the corrected standard error.
  • You can get the corrected p-values by using the following:

=TDIST(ABS(tstat from step10), n-regression df, 2)

ABS is the absolute value of the t-stat from step 10

n-regressiondf will be the same number used in step 5.

  • represents a two-tail test

 

Hausman test for endogeneity

After all this is done, we still need to determine if the variable was endogenous. We employ the Hausman test as a way to check this.

  • First return the first stage regression and obtain the errors (residuals).
  • Thenregress the dependent variable on all the independent variables not including the instr. variables.
  • Then look at the t-stat on the errorvariable.
  • If it is significant, then the variable isendogenous and doing a two-stage model wasappropriate.

 

 

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