Two-stage least squares (TSLS) is a special case of instrumental variables regression. As the name suggests, there are two distinct stages in two-stage least squares. In the first stage, TSLS finds the portions of the endogenous and exogenous variables that can be attributed to the instruments. This stage involves estimating an OLS regression of each variable in the model on the set of instruments. The second stage is a regression of the original equation, with all of the variables replaced by the fitted values from the first-stage regressions. The coefficients of this regression are the TSLS estimates.

Then the coefficients computed in two-stage least squares are given by,

and the standard estimated covariance matrix of these coefficients may be computed using:

To estimate an equation using Two-stage Least Squares, open the equation specification box by choosing Object/New Object.../Equation… or Quick/Estimate Equation… Choose TSLS from the Method: dropdown menu and the dialog will change to include an edit window where you will list the instruments.

Alternately, type the tsls keyword in the command window and hit ENTER.

In the Equation specification edit box, specify your dependent variable and independent variables and enter a list of instruments in the Instrument list edit box.

There are a few things to keep in mind as you enter your instruments:

To illustrate the estimation of two-stage least squares, we use an example from Stock and Watson 2007 (p. 438), which estimates the demand for cigarettes in the United States in 1995. (The data are available in the workfile “Sw_cig.WF1”.) The dependent variable is the per capita log of packs sold LOG(PACKPC). The exogenous variables are a constant, C, and the log of real per capita state income LOG(PERINC). The endogenous variable is the log of real after tax price per pack LOG(RAVGPRC). The additional instruments are average state sales tax RTAXSO, and cigarette specific taxes RTAXS. Stock and Watson use the White covariance estimator for the standard errors.

The equation specification is then,

and the instrument list is:

This specification satisfies the order condition for identification, which requires that there are at least as many instruments (four) as there are coefficients (three) in the equation specification. Note that listing C as an instrument is redundant, since by default, EViews automatically adds it to the instrument list.

To specify the use of White heteroskedasticity robust standard errors, we will select White in the Coefficient covariance matrix dropdown menu on the Options tab. By default, EViews will estimate the using the Ordinary method with d.f. Adjustment as specified in Equation (23.3).

Below we show the output from a regression of LOG(PACKPC) on a constant and LOG(RAVGPRS) and LOG(PERINC), with instrument list “LOG(PERINC) RTAXSO RTAXS”.

EViews identifies the estimation procedure, as well as the list of instruments in the header. This information is followed by the usual coefficient, t-statistics, and asymptotic p-values.

The summary statistics reported at the bottom of the table are computed using the formulae outlined in “Summary Statistics”. Bear in mind that all reported statistics are only asymptotically valid. For a discussion of the finite sample properties of TSLS, see Johnston and DiNardo (1997, p. 355–358) or Davidson and MacKinnon (1993, p. 221–224).

Three other summary statistics are reported: “Instrument rank”, the “J-statistic” and the “Prob(J-statistic)”. The Instrument rank is simply the rank of the instrument matrix, and is equal to the number of instruments used in estimation. The J-statistic is calculated as:

You can adjust your TSLS estimates to account for serial correlation by adding AR terms to your equation specification. EViews will automatically transform the model to a nonlinear least squares problem, and estimate the model using instrumental variables. Details of this procedure may be found in Fair (1984, p. 210–214). The output from TSLS with an AR(1) specification using the default settings with a tighter convergence tolerance looks as follows:

The Options button in the estimation box may be used to change the iteration limit and convergence criterion for the nonlinear instrumental variables procedure.

Suppose your specification is:

EViews estimates the model as a nonlinear regression model so that Fair’s warning does not apply. Estimation of the model does, however, require specification of additional instruments to satisfy the instrument order condition for the transformed specification. By default, the first-stage instruments employed in TSLS are formed as if one were running Cochrane-Orcutt using Fair’s prescription. Thus, if you omit the lagged left- and right-hand side terms from the instrument list, EViews will, by default, automatically add the lagged terms as instruments. This addition will be noted in your output.

You may instead instruct EViews not to add the lagged left- and right-hand side terms as instruments. In this case, you are responsible for adding sufficient instruments to ensure the order condition is satisfied.

The AR(1) results extend naturally to specifications involving higher order serial correlation. For example, if you include a single AR(4) term in your model, the natural instrument list will be:

If you include AR terms from 1 through 4, one possible instrument list is:

Note that while conceptually valid, this instrument list has a large number of overidentifying instruments, which may lead to computational difficulties and large finite sample biases (Fair (1984, p. 214), Davidson and MacKinnon (1993, p. 222-224)). In theory, adding instruments should always improve your estimates, but as a practical matter this may not be so in small samples.

In this case, you may wish to turn off the automatic lag instrument addition and handle the additional instrument specification directly.

Suppose that you wish to estimate the consumption function by two-stage least squares, allowing for first-order serial correlation. You may then use two-stage least squares with the variable list,

and instrument list:

Notice that the lags of both the dependent and endogenous variables (CONS(–1) and GDP(–1)), are included in the instrument list.

Similarly, consider the consumption function:

A valid instrument list is given by:

Here we treat the lagged left and right-hand side variables from the original specification as predetermined and add the lagged values to the instrument list.

Lastly, consider the specification:

Adding all of the relevant instruments in the list, we have:

You can also estimate two-stage least squares variable problems with MA error terms of various orders. To account for the presence of MA errors, simply add the appropriate terms to your specification prior to estimation.

Suppose that you wish to estimate the consumption function by two-stage least squares, accounting for first-order moving average errors. You may then use two-stage least squares with the variable list,

and instrument list:

EViews will add both first and second lags of CONS and GDP to the instrument list.

Most of the technical details are identical to those outlined above for AR errors. EViews transforms the model that is nonlinear in parameters (employing backcasting, if appropriate) and then estimates the model using nonlinear instrumental variables techniques.

Recall that by default, EViews augments the instrument list by adding lagged dependent and regressor variables corresponding to the AR lags. Note however, that each MA term involves an infinite number of AR terms. Clearly, it is impossible to add an infinite number of lags to the instrument list, so that EViews performs an ad hoc approximation by adding a truncated set of instruments involving the MA order and an additional lag. If for example, you have an MA(5), EViews will add lagged instruments corresponding to lags 5 and 6.

Of course, you may instruct EViews not to add the extra instruments. In this case, you are responsible for adding enough instruments to ensure the instrument order condition is satisfied.