First-difference estimator

<h2 id="derivation">Derivation</h2>
The FD estimator avoids bias due to some unobserved, time-invariant variable 
 
 
 
 
 c
 
 i
 
 
 
 
 {\displaystyle c_{i}}
 
, using the repeated observations over time:

y
          
            i
            t
          
        
        =
        
          x
          
            i
            t
          
        
        β
        +
        
          c
          
            i
          
        
        +
        
          u
          
            i
            t
          
        
        ,
        t
        =
        1
        ,
        .
        .
        .
        T
        ,
      
    
    {\displaystyle y_{it}=x_{it}\beta +c_{i}+u_{it},t=1,...T,}

y
          
            i
            t
            −
            1
          
        
        =
        
          x
          
            i
            t
            −
            1
          
        
        β
        +
        
          c
          
            i
          
        
        +
        
          u
          
            i
            t
            −
            1
          
        
        ,
        t
        =
        2
        ,
        .
        .
        .
        T
        .
      
    
    {\displaystyle y_{it-1}=x_{it-1}\beta +c_{i}+u_{it-1},t=2,...T.}

Differencing the equations, gives:

Δ
        
          y
          
            i
            t
          
        
        =
        
          y
          
            i
            t
          
        
        −
        
          y
          
            i
            t
            −
            1
          
        
        =
        Δ
        
          x
          
            i
            t
          
        
        β
        +
        Δ
        
          u
          
            i
            t
          
        
        ,
        t
        =
        2
        ,
        .
        .
        .
        T
        ,
      
    
    {\displaystyle \Delta y_{it}=y_{it}-y_{it-1}=\Delta x_{it}\beta +\Delta u_{it},t=2,...T,}

which removes the unobserved 
 
 
 
 
 c
 
 i
 
 
 
 
 {\displaystyle c_{i}}
 
 and eliminates the first time period.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a><a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>
The FD estimator 
 
 
 
 
 
 
 
 β
 ^
 
 
 
 
 F
 D
 
 
 
 
 {\displaystyle {\hat {\beta }}_{FD}}
 
 is then obtained by using the differenced terms for 
 
 
 
 x
 
 
 {\displaystyle x}
 
 and 
 
 
 
 u
 
 
 {\displaystyle u}
 
 in OLS:

β
                ^
              
            
          
          
            F
            D
          
        
        =
        (
        Δ
        
          X
          ′
        
        Δ
        X
        
          )
          
            −
            1
          
        
        Δ
        
          X
          ′
        
        Δ
        y
        =
        β
        +
        (
        Δ
        
          X
          ′
        
        Δ
        X
        
          )
          
            −
            1
          
        
        Δ
        
          X
          ′
        
        Δ
        u
      
    
    {\displaystyle {\hat {\beta }}_{FD}=(\Delta X'\Delta X)^{-1}\Delta X'\Delta y=\beta +(\Delta X'\Delta X)^{-1}\Delta X'\Delta u}

where 
 
 
 
 X
 ,
 y
 ,
 
 
 {\displaystyle X,y,}
 
 and 
 
 
 
 u
 
 
 {\displaystyle u}
 
, are notation for matrices of relevant variables. Note that the <a href="/facts/Multicollinearity/bQXzB6sA">rank condition</a> must be met for 
 
 
 
 Δ
 
 X
 ′
 
 Δ
 X
 
 
 {\displaystyle \Delta X'\Delta X}
 
 to be invertible (
 
 
 
 
 rank
 
 [
 Δ
 
 X
 ′
 
 Δ
 X
 ]
 =
 k
 
 
 {\displaystyle {\text{rank}}[\Delta X'\Delta X]=k}
 
), where 
 
 
 
 k
 
 
 {\displaystyle k}
 
 is the number of regressors.
Let

Δ
 
 X
 
 i
 
 
 =
 [
 Δ
 
 X
 
 i
 2
 
 
 ,
 Δ
 
 X
 
 i
 3
 
 
 ,
 .
 .
 .
 ,
 Δ
 
 X
 
 i
 T
 
 
 ]
 
 
 {\displaystyle \Delta X_{i}=[\Delta X_{i2},\Delta X_{i3},...,\Delta X_{iT}]}
 
,
and, analogously,

Δ
 
 u
 
 i
 
 
 =
 [
 Δ
 
 u
 
 i
 2
 
 
 ,
 Δ
 
 u
 
 i
 3
 
 
 ,
 .
 .
 .
 ,
 Δ
 
 u
 
 i
 T
 
 
 ]
 
 
 {\displaystyle \Delta u_{i}=[\Delta u_{i2},\Delta u_{i3},...,\Delta u_{iT}]}
 
.
If the error term is strictly exogenous, i.e. 
 
 
 
 E
 [
 
 u
 
 i
 t
 
 
 
 |
 
 
 x
 
 i
 1
 
 
 ,
 
 x
 
 i
 2
 
 
 ,
 .
 .
 ,
 
 x
 
 i
 T
 
 
 ]
 =
 0
 
 
 {\displaystyle E[u_{it}|x_{i1},x_{i2},..,x_{iT}]=0}
 
, by the <a href="/facts/Central_limit_theorem/0gc2L0Wd">central limit theorem</a>, the <a href="/facts/Law_of_large_numbers/X3Bjcy3v">law of large numbers</a>, and the <a href="/facts/Slutsky%2527s_theorem/Fwh5f8Wt">Slutsky's theorem</a>, the estimator is distributed normally with asymptotic variance of

Avar
 ^
 
 
 
 (
 
 
 
 
 β
 ^
 
 
 
 
 F
 D
 
 
 )
 =
 E
 [
 Δ
 
 X
 
 i
 
 ′
 
 Δ
 
 X
 
 i
 
 
 
 ]
 
 −
 1
 
 
 E
 [
 Δ
 
 X
 
 i
 
 ′
 
 Δ
 
 u
 
 i
 
 
 Δ
 
 u
 
 i
 
 ′
 
 Δ
 
 X
 
 i
 
 
 ]
 E
 [
 Δ
 
 X
 
 i
 
 ′
 
 Δ
 
 X
 
 i
 
 
 
 ]
 
 −
 1
 
 
 
 
 {\displaystyle {\widehat {\text{Avar}}}({\hat {\beta }}_{FD})=E[\Delta X_{i}'\Delta X_{i}]^{-1}E[\Delta X_{i}'\Delta u_{i}\Delta u_{i}'\Delta X_{i}]E[\Delta X_{i}'\Delta X_{i}]^{-1}}
 
.
Under the assumption of homoskedasticity and no serial correlation, 
 
 
 
 
 Var
 
 (
 Δ
 u
 
 |
 
 X
 )
 =
 
 σ
 
 Δ
 u
 
 
 2
 
 
 
 
 {\displaystyle {\text{Var}}(\Delta u|X)=\sigma _{\Delta u}^{2}}
 
, the asymptotic variance can be estimated as

Avar
              ^
            
          
        
        (
        
          
            
              
                β
                ^
              
            
          
          
            F
            D
          
        
        )
        =
        
          
            
              
                σ
                ^
              
            
          
          
            Δ
            u
          
          
            2
          
        
        (
        Δ
        
          X
          ′
        
        Δ
        X
        
          )
          
            −
            1
          
        
        ,
      
    
    {\displaystyle {\widehat {\text{Avar}}}({\hat {\beta }}_{FD})={\hat {\sigma }}_{\Delta u}^{2}(\Delta X'\Delta X)^{-1},}

where 
 
 
 
 
 
 
 
 σ
 ^
 
 
 
 
 u
 
 
 2
 
 
 
 
 {\displaystyle {\hat {\sigma }}_{u}^{2}}
 
, a consistent estimator of 
 
 
 
 
 σ
 
 u
 
 
 2
 
 
 
 
 {\displaystyle \sigma _{u}^{2}}
 
, is given by

σ
                ^
              
            
          
          
            Δ
            u
          
          
            2
          
        
        =
        [
        n
        (
        T
        −
        1
        )
        −
        K
        
          ]
          
            −
            1
          
        
        
          ∑
          
            i
            =
            1
          
          
            n
          
        
        
          ∑
          
            t
            =
            2
          
          
            T
          
        
        
          
            
              
                
                  Δ
                  
                    u
                    
                      i
                      t
                    
                  
                
                ^
              
            
          
          
            2
          
        
      
    
    {\displaystyle {\hat {\sigma }}_{\Delta u}^{2}=[n(T-1)-K]^{-1}\sum _{i=1}^{n}\sum _{t=2}^{T}{\widehat {\Delta u_{it}}}^{2}}

and

Δ
 
 u
 
 i
 t
 
 
 
 ^
 
 
 
 =
 Δ
 
 y
 
 i
 t
 
 
 −
 
 
 
 
 β
 ^
 
 
 
 
 F
 D
 
 
 Δ
 
 x
 
 i
 t
 
 
 
 
 {\displaystyle {\widehat {\Delta u_{it}}}=\Delta y_{it}-{\hat {\beta }}_{FD}\Delta x_{it}}
 
.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>
<h2 id="properties">Properties</h2>
To be unbiased, the fixed effects estimator (FE) requires strict exogeneity, defined as

E
 [
 
 u
 
 i
 t
 
 
 
 |
 
 
 x
 
 i
 1
 
 
 ,
 
 x
 
 i
 2
 
 
 ,
 .
 .
 ,
 
 x
 
 i
 T
 
 
 ]
 =
 0
 
 
 {\displaystyle E[u_{it}|x_{i1},x_{i2},..,x_{iT}]=0}
 
.
The first difference estimator (FD) is also unbiased under this assumption.
If strict exogeneity is violated, but the weaker assumption

E
        [
        (
        
          u
          
            i
            t
          
        
        −
        
          u
          
            i
            t
            −
            1
          
        
        )
        (
        
          x
          
            i
            t
          
        
        −
        
          x
          
            i
            t
            −
            1
          
        
        )
        ]
        =
        0
      
    
    {\displaystyle E[(u_{it}-u_{it-1})(x_{it}-x_{it-1})]=0}

holds, then the FD estimator is consistent.
Note that this assumption is less restrictive than the assumption of strict exogeneity which is required for consistency using the FE estimator when 
 
 
 
 T
 
 
 {\displaystyle T}
 
 is fixed. If 
 
 
 
 T
 →
 ∞
 
 
 {\displaystyle T\rightarrow \infty }
 
, then both FE and FD are consistent under the weaker assumption of contemporaneous exogeneity.
The <a href="/facts/Hausman_test/zsNFBvBS">Hausman test</a> can be used to test the assumptions underlying the consistency of the FE and FD estimators.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>

<h2 id="relation-to-fixed-effects-estimator">Relation to fixed effects estimator</h2>
For 
 
 
 
 T
 =
 2
 
 
 {\displaystyle T=2}
 
, the FD and fixed effects estimators are numerically equivalent.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>
Under the assumption of <a href="/facts/Homoscedasticity/50v2qyEN">homoscedasticity</a> and no <a href="/facts/Serial_correlation/dm4MqK0A">serial correlation</a> in 
 
 
 
 
 u
 
 i
 t
 
 
 
 
 {\displaystyle u_{it}}
 
, the FE estimator is more <a href="/facts/Efficiency_(statistics)/wTmgEqte">efficient</a> than the FD estimator. This is because the FD estimator induces no serial correlation when differencing the errors. If 
 
 
 
 
 u
 
 i
 t
 
 
 
 
 {\displaystyle u_{it}}
 
 follows a <a href="/facts/Random_walk/08v0jmfv">random walk</a>, however, the FD estimator is more efficient as 
 
 
 
 Δ
 
 u
 
 i
 t
 
 
 
 
 {\displaystyle \Delta u_{it}}
 
 are serially uncorrelated.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Factor_analysis/LT6B9I7D">Factor analysis</a></li>
<li><a href="/facts/Panel_analysis/lk2LMm6b">Panel analysis</a></li>
<li><a href="/facts/Fixed_effect_model/L5JPDQnB">Fixed effect model</a></li></ul>
<h2 id="notes">Notes</h2>

<ul><li>Wooldridge, Jeffrey M. (2001). <a href="https://archive.org/details/econometricanaly0000wool">Econometric Analysis of Cross Section and Panel Data</a>. MIT Press. pp. <a href="https://archive.org/details/econometricanaly0000wool/page/279">279</a>–291. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-262-23219-7. Retrieved 30 August 2024.</li></ul>
<ul><li>Wooldridge, Jeffrey M. (2013). <a href="https://cbpbu.ac.in/userfiles/file/2020/STUDY_MAT/ECO/2.pdf">Introductory Econometrics: A Modern Approach</a> (PDF) (5th ed.). South-Western Cengage Learning. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-111-53104-1. Retrieved 30 August 2024.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Wooldridge 2001, p. 284. <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Wooldridge 2013, p. 461. <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Wooldridge 2001, p. 279. <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Wooldridge 2001, p. 281. <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Wooldridge 2001, p. 285. <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Wooldridge 2001, p. 284. <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Wooldridge 2001, p. 284. <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
</ol>

First-difference estimator open-in-new

First-difference estimator