<p>It is allowed to use lags. In fact using  using lags as instruments is actually quite common in some fields like macroeconomics (See Romer Advanced Macroeconomics pp 376).</p>
<p>However, note:</p>
<p><span class="math-container">$$
\Delta y_{it} = \beta_1 \Delta y_{it-1} + \beta_2 \Delta b_{it} +\epsilon_{it}
$$</span></p>
<p>Is the reduced form of IV so it will give you estimate of: <span class="math-container">$\beta_1 \pi$</span>, instead of just <span class="math-container">$\beta_1$</span> but you are getting rid of endogeneity. If you want to get just <span class="math-container">$\beta_1$</span> you could run 2SLS.</p>
<p>This is because you are basically substituting <span class="math-container">$a_{it} = \pi_0 + \pi y_{it-1} +e_{it}$</span> so:</p>
<p><span class="math-container">$$
\Delta y_{it} = \beta_1 \pi_0 +  \beta_1  \pi y_{it-1}  +  \beta_2 \Delta b_{it} + \beta_1 e_{it} +\epsilon_{it}
$$</span></p>
<p>You should not get an error doing that using common packages/programs.</p>
<p>If you are trying to program the IV estimator yourself in something like R you are likely getting  the error because when you create matrix containing <span class="math-container">$\Delta y_{t-1}$</span> it won't match the matrix of <span class="math-container">$\Delta y_t$</span> as there is no way of having <span class="math-container">$\Delta y_{t-1}$</span> for the first row. A simple solution to this would be just to exclude the first row of your data matrix after you create <span class="math-container">$\Delta y_{t-1}$</span> variable from regression (or you could even outright delete it but I presume that data can still be useful for some summary statistics/visualizations).</p>

CC BY-SA 4.0
2022-02-26T18:06:17.000Z
2022-02-26T18:19:12.477Z
2022-02-26T18:19:12.477Z
1muflon1
<p>Say, you have a model
<span class="math-container">$$
y_{it} =  \beta_1 a_{it} + \beta_2 b_{it}
$$</span>
And you find that <span class="math-container">$a_{it}$</span> is endogenous, therefore, you need to find an instrument for it.</p>
<p>Say you find an instrument <span class="math-container">$z_{it} = \Delta y_{it-1}$</span> that is relevant and exogenous,</p>
<p>Would you be able to estimate <span class="math-container">$\beta_1$</span> and <span class="math-container">$\beta_2$</span> by estimating</p>
<p><span class="math-container">$$
\Delta y_{it} = \beta_1 \Delta y_{it-1} + \beta_2 \Delta b_{it}
$$</span></p>
<p>And if so, how would this be done? I am running into a problem where the matrix of <span class="math-container">$\Delta y_{it-1}$</span> has fewer rows than <span class="math-container">$\Delta b_{it}$</span>.</p>
<p>Would you just reshape <span class="math-container">$\Delta b_{it}$</span>? Is this allowed?</p>

<econometrics><regression><instrumental-variable>
Estimating parameters when introducing IV Instrument
Zebraboard
# Estimating parameters when introducing IV Instrument

(Asked by: username_0 on 2022-02-26)

Say, you have a model
$$
y\_{it} = \beta\_1 a\_{it} + \beta\_2 b\_{it}
$$
And you find that $a\_{it}$ is endogenous, therefore, you need to find an instrument for it.

Say you find an instrument $z\_{it} = \Delta y\_{it-1}$ that is relevant and exogenous,

Would you be able to estimate $\beta\_1$ and $\beta\_2$ by estimating

$$
\Delta y\_{it} = \beta\_1 \Delta y\_{it-1} + \beta\_2 \Delta b\_{it}
$$

And if so, how would this be done? I am running into a problem where the matrix of $\Delta y\_{it-1}$ has fewer rows than $\Delta b\_{it}$.

Would you just reshape $\Delta b\_{it}$? Is this allowed?

**Tags:** econometrics, regression, instrumental-variable

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## 1 answers

### Answer by username_1 on 2022-02-26. Score: 1 | Accepted answer

It is allowed to use lags. In fact using using lags as instruments is actually quite common in some fields like macroeconomics (See Romer Advanced Macroeconomics pp 376).

However, note:

$$
\Delta y\_{it} = \beta\_1 \Delta y\_{it-1} + \beta\_2 \Delta b\_{it} +\epsilon\_{it}
$$

Is the reduced form of IV so it will give you estimate of: $\beta\_1 \pi$, instead of just $\beta\_1$ but you are getting rid of endogeneity. If you want to get just $\beta\_1$ you could run 2SLS.

This is because you are basically substituting $a\_{it} = \pi\_0 + \pi y\_{it-1} +e\_{it}$ so:

$$
\Delta y\_{it} = \beta\_1 \pi\_0 + \beta\_1 \pi y\_{it-1} + \beta\_2 \Delta b\_{it} + \beta\_1 e\_{it} +\epsilon\_{it}
$$

You should not get an error doing that using common packages/programs.

If you are trying to program the IV estimator yourself in something like R you are likely getting the error because when you create matrix containing $\Delta y\_{t-1}$ it won't match the matrix of $\Delta y\_t$ as there is no way of having $\Delta y\_{t-1}$ for the first row. A simple solution to this would be just to exclude the first row of your data matrix after you create $\Delta y\_{t-1}$ variable from regression (or you could even outright delete it but I presume that data can still be useful for some summary statistics/visualizations).


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