Introduction to Dynamic Panel Data:

Autoregressive Models with Fixed Effects

Eric Zivot

Winter, 2013

Dynamic Panel Data (without covariates)

= −1 + + = 1 (individuals)

= 1 (time periods)

Typical assumptions

1. Stationarity: || 1

2. [|0 −1 ] = 0

3a. No serial correlation: ∼ (0 2)

3b. Homoskedasticity: ∼ (0 2)

Stationary Model Representation

= −1 + +

By recursive substitution

1 = 0 + + 1

2 = 1 + + 2

= [0 + + 1] + + 2

= 20 + (1 + ) + 1 + 2...

= 0 +

⎛⎝−1X=0

⎞⎠+−1X=0

−

Now

[|] = [0|] +

⎛⎝−1X=0

⎞⎠+

−1X=0

[−|]

= [0|] +

⎛⎝−1X=0

⎞⎠For large

≈ 0−1X=0

≈ 1

1−

so that (for large )

[|] ≈1−

var[|] ≈∞X=0

var[−|]

=2

1− 2

Estimation

= −1 + +

= 0 +

⎛⎝−1X=0

⎞⎠+−1X=0

−

Can’t use RE estimation because

[−1 · ]

=

⎡⎣⎛⎝−10 +

⎛⎝−2X=0

⎞⎠+−2X=0

−

⎞⎠

⎤⎦ 6= 0

What about FE estimation?

Write the model in FE matrix notation as

y = y−1 + 1 + η

y×1

=

⎡⎢⎣ 1...

⎤⎥⎦ y−1×1

=

⎡⎢⎣ 0...

−1

⎤⎥⎦ η =⎡⎢⎣ 1...

⎤⎥⎦Define

Q = I −P1so that

Qy = y =

⎛⎜⎝ 1 − ...

−

⎞⎟⎠ =1

X=1

Then, the transformed model is

Qy = Qy−1 + Q1 +Qη⇒y = y−1 + η

The FE estimator of is the pooled OLS estimator on the transformed model

=

⎛⎝ X=1

y0−1y−1

⎞⎠−1 X=1

y0−1y

=

⎛⎝ X=1

y0−1Qy−1

⎞⎠−1 X=1

y0−1Qy

For consistency, consider

− =

⎛⎝1

X=1

y0−1Qy−1

⎞⎠−1 1

X=1

y0−1Qη

will be consistent if

lim→∞

1

X=1

y0−1Qη = [y0−1Qη] = 0

Note: In the asymptotic analysis, the cross section dimension →∞ but thetime series dimension is held fixed!

Now

y0−1Qη =h0 −1

i⎡⎢⎣ 1 − ... −

⎤⎥⎦=

X=1

−1( − )

=1

X=1

−1 = −10 +

⎛⎝−2X=0

⎞⎠+−2X=0

−

Using

−1 = −10 +

⎛⎝−2X=0

⎞⎠+−2X=0

−

it follows that

[y0−1Qη] =X

=1

h−1( − )

i6= 0

because

[−1] =

⎡⎣−1⎛⎝ 1

X=1

⎞⎠⎤⎦ 6= 0

Result:

9 as →∞ for fixed

due to correlation between −1 and

Remark:

If both →∞ and →∞ then it can be shown that

→

because

=1

X=1

→ 0⇒ [−1] = 0

This is an example of “double asymptotic analysis” that is common in theanalysis of panel data models.

Bias of Fixed Effects Estimator

In the stationary model with ∼ iid (0 2) Nickell (1981, Ecta) showedthat for fixed and

[y0−1Qη] = −2()

() =1

1−

"1− 1

Ã1−

1−

!#⇒ is downward biased

Further, Nickell showed that for fixed as →∞

− → −(1− 2)

− 1

Ã1− 2()

− 1

!Notice that as →∞ and →∞ sequentially

− → 0

Example of Bias in FE estimation of dynamic panel model

FE bias: − / .05 .5 .952 -0.52 -0.75 -0.973 -0.35 -0.54 -0.7310 -0.11 -0.16 -0.2615 -0.07 -0.11 -0.17

Remarks

1. If 0 the bias is always negative, and massive for very small values of

2. As increases, the bias decreases but even with = 15 the bias is stillsubstantial for large .

IV Estimation

Anderson and Hsiao (1981) suggested the following approach

1. First eliminate the fixed effect by taking first differences

− −1 = (−1 − −2) + − −1∆ = ∆−1 +∆ = 2

Note

[∆−1∆] 6= 0

due to correlation between −1 and −1 terms.

Stacking across time gives

∆y(−1)×1

= ∆y−1 +∆η

2. Do IV estimation using −2 as an instrument for ∆−1

[−2∆−1] = [−2(−1 − −2)] 6= 0[−2∆] = [−2

³ − −1

´] = 0

since is iid.

The IV estimator is then

=

⎛⎝ X=1

y0−2∆y−1

⎞⎠−1 X=1

y0−2∆y

=

P=1

P=2 −2∆P

=1P=2 −2∆−1

which is consistent by construction.

Remark:

The Anderson-Hsiao estimator does not exploit all the relevant moment condi-tions so it is not the most efficient GMM estimator.

Arellano-Bond GMM Estimator

“Tests of Specification for Panel Data: Monte Carlo Evidence and an Applica-tion to Employment Equations”, Review of Economic Studies, 58, 1991

Arellano and Bond (AB) derived all of the relevant moment conditions fromthe dynamic panel data model to be used in GMM estimation.

The moment condtions are based on the first differenced model

∆ = ∆−1 +∆ = 2

They showed that the number of moment conditions depends on (numberof time periods)

Example: = 4⇒ 6 moment condtions

= 2 : [∆20] = 0

= 3 :[∆30] = 0[∆31] = 0

= 4 :[∆40] = 0[∆41] = 0[∆42] = 0

For GMM estimation, define

g4 () =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

∆20∆30∆31∆40∆41∆42

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

(∆2 − ∆1) 0(∆3 − ∆3) 0(∆3 − ∆2) 1(∆4 − ∆3) 0(∆4 − ∆3) 1(∆4 − ∆3) 2

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

Notice that g4 () is a linear function of It may be re-expressed in matrixform as

g4 () =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0 00 0 00 1 00 0 00 0 10 0 2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦⎡⎢⎣ ∆2 − ∆1∆3 − ∆3∆4 − ∆3

⎤⎥⎦

= X40 [∆y4 − ∆y4−1]

where

∆y4 =

⎡⎢⎣ ∆2∆3∆4

⎤⎥⎦ ∆y4−1 =

⎡⎢⎣ ∆1∆2∆3

⎤⎥⎦X4 =

⎡⎢⎣ 0 0 0 0 0 00 0 1 0 0 00 0 0 0 1 2

⎤⎥⎦

The sample moments used for GMM estimation are then

g4() =1

X=1

X40 [∆y4 − ∆y4−1]

= S4∆ − S4∆−1

S4∆ =1

X=1

X40 ∆y4

S4∆−1 =1

X=1

X40 ∆y4−1

Because is randomly sampled over it follows that

S6×6

= [g4 ()g4 ()

0] = [X40 ∆η∆η0X4 ]

Under conditional heteroskedasticity, a consistent estimate is

S =1

X=1

X40 ∆η∆η0X4

∆η = [∆y4 − ∆y4−1]

→

For example, can use the Anderson and Hsiao IV estimate of

Under conditional homoskedasticity

S = [X40 ∆η∆η0X4 ] = [[X40 C

0ηη0CX

4 |X4 ]] =

= [X40 C0[ηη

0|X4 ]CX4 ] = 2[X

40 C

0CX4 ]

where

C0(−1)×

=

⎛⎜⎜⎜⎝−1 1 0 00 −1 1 0

0 0 0 −1 1

⎞⎟⎟⎟⎠So a consistent estimate of S is

S =1

X=1

X40 C0CX4

Estimation of 2 is not required for GMM estimation because it cancels out inthe resulting estimator.

The efficient GMM estimator solves

min

( S−1) = g4()S−1g4()

= [S4∆ − S4∆−1]0S−1[S4∆ − S4∆−1]

Since ( S−1) is linear in the analytic solution is

(S−1) = (S40∆−1S

−1S4∆−1)S40∆−1S

−1S4∆

This estimator is known as the Arellano-Bond GMM estimator.

Example: International Difference in Output Growth Rates (Hayashi,Section 5.4)

Q: Do poor countries grow faster than rich countries? If so how much faster?

Neoclassical growth theory foundations

() = output per effective labor at time for a country

→ ∗ = steady state

The log-linear approximation around ∗ gives the adjustment equation

ln(())

= · [ln(∗)− ln(()]

= speed of convergence 0

Between any two time periods −1 and , the log-linear adjustment equationimplies

ln(()) = (1− ) · ln(∗) + ln((−1))

= exp [− · ( − −1)]

Define

() = ()

()() () = aggregate output

() = aggregate hours worked

() = level of labor augmenting technical progress

Assume () grows at a constant rate Then () = (0) exp( · ) and

ln () = ln

à ()

()

!− ln((0))− ·

Substituting into the output equation gives

ln

à ()

()

!= ln

à (−1)(−1)

!+ (1− ) [ln(∗)− ln((0))] +

= · ( − · −1)

Subtracting lnµ (−1)(−1)

¶from both sides gives the growth equation

∆ ln

à ()

()

!= (− 1) ln

à (−1)(−1)

!+(1− ) [ln(∗)− ln((0)] +

Because 1, the level of per capita output has a negative effect on growth.Hence, poor countries should grow faster than rich countries.

Turning the output equation into a dynamic panel data model

Assume

ln

à ()

()

!= ln

à (−1)(−1)

!+ (1− ) [ln(∗)− ln((0)] +

= · ( − · −1)

holds for every country where and are the same for every country but that∗ and might differ. Then we can write

= + −1 + +

= ln ( ()()) = log per capita output at time = (1− ){ln(∗ )− ln((0))} for country

= country and time specific shock (e.g. business cycle)