Testing

[Enrique Pinzon (StataCorp)]

\begin{equation}
y_{it} = x_{it}'\beta_1 + y_{i(t-1)}\beta_2 + \alpha_i + \varepsilon_{it}
\end{equation}

As in the conventional linear panel data model the time-invariant unobserved component is related to the regressors but the strategy of taking first differences does not work because of an endogeneity problem. In particular:

\begin{eqnarray}
\Delta y_{it} = \Delta x_{it}'\beta_1 + \Delta y_{i(t-1)} + \Delta \varepsilon_{it}
\end{eqnarray}

[Enrique Pinzon (StataCorp)]

\begin{eqnarray}
\Delta y_{it} &=& \Delta x_{it}'\beta_1 + \Delta y_{i(t-1)} + \Delta \varepsilon_{it} \\
E\left( \Delta y_{i(t-1)} \Delta \varepsilon_{it} \right) &\neq & 0
\end{eqnarray}

[Enrique Pinzon (StataCorp)]

\begin{eqnarray}
\Delta y_{it} &=& \Delta x_{it}'\beta_1 + \Delta y_{i(t-1)} + \Delta \varepsilon_{it} \\
E\left( \Delta y_{i(t-1)} \Delta \varepsilon_{it} \right) & \neq & 0
\end{eqnarray}

[Enrique Pinzon (StataCorp)]

\begin{eqnarray}
\Delta y_{it} &=& \Delta x_{it}'\beta_1 + \Delta y_{i(t-1)} + \Delta \varepsilon_{it} \\
E\left( \Delta y_{i(t-1)} \Delta \varepsilon_{it} \right) & = & 0
\end{eqnarray}

[Enrique Pinzon (StataCorp)]

\begin{equation}
\Delta \varepsilon_{it} = \Delat \nu_{it}
\end{equation}

But we are in trouble if:

\begin{equation}
\Delta \varepsilon_{it} = \Delta \nu_{it} + \Delta \nu_{i(t-1)}
\end{equation}

[Enrique Pinzon (StataCorp)]

Hello Philipp,

Jorge Eduardo is correct in pointing out that you have a dynamic panel data model. You could use an Arellano-Bond xtabond specification or you could try using xtdpd which allows you to have different correlation structures for the unobserved time-invariant components of your model. Arellano-Bond is very popular but one has to be careful to verify that the assumptions of the model are satisfied. The documentation is a great source of information to get acquainted with the assumptions and the tests on this subject. That being said, I also want to provide answers to some frequently asked questions:

1. xtabond and xtdpd use GMM to estimate the parameters of the model. It is common for people to ask where the instruments are coming from and what the intuition behind them is. This is what I have to say:

The relationship of interest is given by:

\begin{equation}
y_{it} = x_{it}'\beta_1 + y_{i(t-1)}\beta_2 + \alpha_i + \varepsilon_{it}
\end{equation}

As in the conventional linear panel data model the time-invariant unobserved component is related to the regressors but the strategy of taking first differences does not work because of an endogeneity problem. In particular:

\begin{eqnarray}
\Delta y_{it} &=& \Delta x_{it}'\beta_1 + \Delta y_{i(t-1)} + \Delta \varepsilon_{it} \\
E\left( \Delta y_{i(t-1)} \Delta \varepsilon_{it} \right) &\neq & 0
\end{eqnarray}

The second equality above defines the endogeneity problem. The solution is instrumental variables. Which instrumental variables? Arellano-Bond suggest the second lags of the dependent variable and all the lags thereafter. This generates the set of moment conditions defined by:

\begin{eqnarray}
E\left( \Delta y_{i(t-2)} \Delta \varepsilon_{it} \right) &=& 0 \\
E\left( \Delta y_{i(t-3)} \Delta \varepsilon_{it} \right) &=& 0 \\
\ldots & &
\end{eqnarray}

If you have 6 time periods then your instrument set will look like :
\begin{eqnarray} t=6 & \quad y_{t-4}, y_{t-3}, y_{t-2}, y_{t-1} \\ t=5 & \quad y_{t-3}, y_{t-2}, y_{t-1} \\ t=4 & \quad y_{t-2}, y_{t-1} \\ t=3 & \quad y_{t-1} \end{eqnarray} Below I show an example with 9 periods which, following the logic above, should give you 28 lagged instruments.

. webuse abdata

. tab year

       year |      Freq.     Percent        Cum.
------------+-----------------------------------
       1976 |         80        7.76        7.76
       1977 |        138       13.39       21.14
       1978 |        140       13.58       34.72
       1979 |        140       13.58       48.30
       1980 |        140       13.58       61.88
       1981 |        140       13.58       75.46
       1982 |        140       13.58       89.04
       1983 |         78        7.57       96.61
       1984 |         35        3.39      100.00
------------+-----------------------------------
      Total |      1,031      100.00

. xtabond n, noconstant

Arellano-Bond dynamic panel-data estimation     Number of obs     =        751
Group variable: id                              Number of groups  =        140
Time variable: year
                                                Obs per group:
                                                              min =          5
                                                              avg =   5.364286
                                                              max =          7

Number of instruments =     28                  Wald chi2(1)      =     283.40
                                                Prob > chi2       =     0.0000
One-step results
------------------------------------------------------------------------------
           n |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
           n |
         L1. |   1.023349   .0607887    16.83   0.000     .9042054    1.142493
------------------------------------------------------------------------------
Instruments for differenced equation
        GMM-type: L(2/.).n

2. How should I interpret my estat abond results ?

The key for the instrument set in Arellano-Bond to work is that

\begin{equation}
E\left( \Delta y_{i(t-j)} \Delta \varepsilon_{it} \right) = 0 \quad j \geq 2
\end{equation}

estat abond is testing these conditions. In essence, the differenced unobserved time-invariant component should be unrelated to the second lag of the dependent variable and the lags thereafter. If this is not the case, we are back to the initial problem, endogeneity. Essentially:

All is well if:

\begin{equation}
\Delta \varepsilon_{it} = \Delta \nu_{it}
\end{equation}

But we are in trouble if:

\begin{equation}
\Delta \varepsilon_{it} = \Delta \nu_{it} + \Delta \nu_{i(t-1)}
\end{equation}

If the latter case is true, then the second lag of the dependent variable will be related to the differenced time-varying component. This happens in general if the differenced time-varying unobserved component is serially correlated with an order greater than one. In the first equation above, the unobservable is serially correlated of order one but not serially correlated of orders 2 or beyond. Therefore, we are fine. This is what estat abond tests. For the example above:

. estat abond

Arellano-Bond test for zero autocorrelation in first-differenced errors
  +-----------------------+
  |Order |  z     Prob > z|
  |------+----------------|
  |   1  |-7.2051  0.0000 |
  |   2  |-1.4815  0.1385 |
  +-----------------------+
   H0: no autocorrelation

We are golden. We reject no autocorrelation of order one and cannot reject no autocorrelation of order 2. If this were not the case, we would have to look for different instruments. This is what xtdpd does. The following example shows a case were the serial correlation structure of Arellano-Bond is not satisfied:

. quietly xtabond cap, nocons

. estat abond, artests(3)

Dynamic panel-data estimation                   Number of obs     =        751
Group variable: id                              Number of groups  =        140
Time variable: year
                                                Obs per group:
                                                              min =          5
                                                              avg =   5.364286
                                                              max =          7

Number of instruments =     28                  Wald chi2(1)      =      30.31
                                                Prob > chi2       =     0.0000
One-step results
------------------------------------------------------------------------------
         cap |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         cap |
         L1. |    .269652   .0489782     5.51   0.000     .1736565    .3656474
------------------------------------------------------------------------------
Instruments for differenced equation
        GMM-type: L(2/.).cap

Arellano-Bond test for zero autocorrelation in first-differenced errors
  +-----------------------+
  |Order |  z     Prob > z|
  |------+----------------|
  |   1  |-4.6252  0.0000 |
  |   2  |-3.3436  0.0008 |
  |   3  | 1.4573  0.1450 |
  +-----------------------+
   H0: no autocorrelation

In this case we reject the no serial correlation for a second order of the differenced unobservables.

[Enrique Pinzon (StataCorp)]

\begin{equation}
y = \beta_0 + \beta_1dm + \beta_2dt + \beta_3 dt \dot dm + \varepsilon
\end{equation}

[Enrique Pinzon (StataCorp)]

\begin{equation}
y = \beta_0 + \beta_1dm + \beta_2dt + \beta_3 dt \times dm + \varepsilon
\end{equation}

[Enrique Pinzon (StataCorp)]

\begin{equation}
y_t = x_t\beta
\end{equation}

[Enrique Pinzon (StataCorp)]

Anne,

Using factor variable notation when you have a nonlinear model like probit is important to get the marginal effects you want. Suppose, the variable x1 is categorical variable with two values and when introducing it in your list of covariates instead of typing

i.x1

you type

x1

The marginal effect ( dydx(x1)) in the first case is

\begin{equation}
\Phi\left(\beta_0 + 1 + \beta_2x_2 \ldots \right) - \Phi\left(\beta_0 + 0 + \beta_2x_2 \ldots \right)
\end{equation}

In the second case it is given by:

\begin{equation}
\phi \left(\beta_0 + \beta_1x1 + \beta_2x_2 + \ldots \right)\beta_1
\end{equation}