Panel Model

• $y_i: T\times 1$, $Y_{nT\times 1}=(y_1',...,y_n')'$ ,
• $x_i : T_i\times k$, $X_{nT\times k}=(x_1',...,x_n')'$

$$\begin{aligned} &y_{i t}=\boldsymbol{x}_{i t}^{\prime} \boldsymbol{\beta}+e_{i t}\\ &Y=X\beta+e\\ &e_{i t}=\alpha_{i}+\varepsilon_{i t} \text{ individual effect+idiosyncratic error} \end{aligned}$$

1. Random Effects-GLS

$y_{i t}=\boldsymbol{x}_{i t}^{\prime} \boldsymbol{\beta}+\alpha_i+\varepsilon_{i t}$ RE-Identification

$$\begin{aligned} \mathbb{E}\left(\varepsilon_{i t} \mid \mathbf{X}_{i}\right) &=0 \\ \mathbb{E}\left(\varepsilon_{i t}^{2} \mid \mathbf{X}_{i}\right) &=\sigma_{\varepsilon}^{2} \\ \mathbb{E}\left(\varepsilon_{i j} \varepsilon_{i t} \mid \mathbf{X}_{i}\right) &=0 \\ \mathbb{E}\left(\alpha_{i} \mid \mathbf{X}_{i}\right) &=0 \\ \mathbb{E}\left(\alpha_{i}^{2} \mid \mathbf{X}_{i}\right) &=\sigma_{\alpha}^{2} \\ \mathbb{E}\left(\alpha_{i} \varepsilon_{i t} \mid \mathbf{X}_{i}\right) &=0 \end{aligned}$$

For that heteroskedasticity,

$$\begin{aligned} \mathbb{E}\left(\boldsymbol{e}_{i} \mid \mathbf{X}_{i}\right) &=0 \\ \mathbb{E}\left(\boldsymbol{e}_{i} \boldsymbol{e}_{i}^{\prime} \mid \mathbf{X}_{i}\right) &=\mathbf{1}_{i} \mathbf{1}_{i}^{\prime} \sigma_{u}^{2}+\boldsymbol{I}_{i} \sigma_{\varepsilon}^{2}=\sigma_{\varepsilon}^{2} \boldsymbol{\Omega}_{i}. \end{aligned}$$

We use GLS to obtain BLUE (Gauss-Markov conditions).

$$\widehat{\boldsymbol{\beta}}_{\mathrm{gls}}=\left(\sum_{i=1}^{N} \boldsymbol{X}_{i}^{\prime} \boldsymbol{\Omega}_{i}^{-1} \boldsymbol{X}_{i}\right)^{-1}\left(\sum_{i=1}^{N} \boldsymbol{X}_{i}^{\prime} \boldsymbol{\Omega}_{i}^{-1} \boldsymbol{y}_{i}\right)$$

Feasible We use still consistent OLS for the residuals $\hat{e}$ and estimate the covariance matrix.

$$\widehat{\boldsymbol{e}}_{i}=\boldsymbol{y}_{i}-\mathbf{X}_{i} \widehat{\boldsymbol{\beta}}_{\mathrm{OLS}}$$

2. Fixed Effect

In the econometrics literature, if the stochastic structure of $\alpha_i$ is treated as unknown and possibly correlated with $x_{it}$ then $\alpha_i$ is called a fixed effect.

FE-Identification

• $E(\varepsilon_{it} | x_{i},\alpha_i)=0$ for all $t$
• $E(\alpha_i |x_i)\neq 0$ check by ==Hausman-Wu test==. otherwise use random effects to achieve more efficiency.
  $$\begin{aligned} H &=\left(\widehat{\boldsymbol{\beta}}_{\mathrm{fe}}-\widehat{\boldsymbol{\beta}}_{\mathrm{re}}\right)^{\prime} \widehat{\operatorname{var}}\left(\widehat{\boldsymbol{\beta}}_{\mathrm{fe}}-\widehat{\boldsymbol{\beta}}_{\mathrm{re}}\right)^{-1}\left(\widehat{\boldsymbol{\beta}}_{\mathrm{fe}}-\widehat{\boldsymbol{\beta}}_{\mathrm{re}}\right) \\ &=\left(\widehat{\boldsymbol{\beta}}_{\mathrm{fe}}-\widehat{\boldsymbol{\beta}}_{\mathrm{re}}\right)^{\prime}\left(\widehat{\boldsymbol{V}}_{\mathrm{fe}}-\widehat{\boldsymbol{V}}_{\mathrm{re}}\right)^{-1}\left(\widehat{\boldsymbol{\beta}}_{\mathrm{fe}}-\widehat{\boldsymbol{\beta}}_{\mathrm{re}}\right) \end{aligned}$$

Within-transformation $M Y=M X\beta+M\alpha +M \varepsilon$

$$\begin{aligned} \widehat{\boldsymbol{\beta}}_{\mathrm{fe}}&=\left(\sum_{i=1}^{N} \sum_{t \in S_{i}} \dot{\boldsymbol{x}}_{i t} \dot{\boldsymbol{x}}_{i t}^{\prime}\right)^{-1}\left(\sum_{i=1}^{N} \sum_{t \in S_{i}} \dot{\boldsymbol{x}}_{i t} \dot{y}_{i t}\right) \\ &=\left(\sum_{i=1}^{N} \dot{\boldsymbol{X}}_{i}^{\prime} \dot{\boldsymbol{X}}_{i}\right)^{-1}\left(\sum_{i=1}^{N} \dot{\boldsymbol{X}}_{i}^{\prime} \dot{\boldsymbol{y}}_{i}\right) \\ &=\left(\sum_{i=1}^{N} \boldsymbol{X}_{i}^{\prime} \mathbf{M}_{i} \mathbf{X}_{i}\right)^{-1}\left(\sum_{i=1}^{N} \boldsymbol{X}_{i}^{\prime} \mathbf{M}_{i} \boldsymbol{y}_{i}\right) \end{aligned}$$

Dummy Regression

Take $\alpha_i$ as coefficient of a dummy variable:

$$\begin{aligned} &d_i=(d_{i1},\cdots,d_{ii},\cdots,d_{in})'=(0,\cdots,1,\cdots,0)'\\ &\alpha=(\alpha_{1},\cdots,\alpha_{i},\cdots,\alpha_{n})' \end{aligned}$$

$y_{it}=x_{it}'\beta +d_i'\alpha +\varepsilon_{it}$ Then by OLS and Frisch-Waugh Theorem, the $\hat{\beta}$ is same as fixed effect estimator.

RE v.s. FE

RE is a linear combination of between estimator $\left(\sum_{i=1}^{N} \boldsymbol{X}_{i}^{\prime} \mathbf{P}_{i} \mathbf{X}_{i}\right)^{-1}\left(\sum_{i=1}^{N} \boldsymbol{X}_{i}^{\prime} \mathbf{P}_{i} \boldsymbol{y}_{i}\right)$ and fixed effect estimator $\left(\sum_{i=1}^{N} \boldsymbol{X}_{i}^{\prime} \mathbf{M}_{i} \mathbf{X}_{i}\right)^{-1}\left(\sum_{i=1}^{N} \boldsymbol{X}_{i}^{\prime} \mathbf{M}_{i} \boldsymbol{y}_{i}\right)$.

$$\bar{y}_{i}=\overline{\boldsymbol{x}}_{i}^{\prime} \boldsymbol{\beta}+\alpha_{i}+\bar{\varepsilon}_{i}, \quad \widehat{\boldsymbol{\beta}}_{\mathrm{be}}=\left(\sum_{i=1}^{N} \overline{\boldsymbol{x}}_{i} \overline{\boldsymbol{x}}_{i}^{\prime}\right)^{-1}\left(\sum_{i=1}^{N} \overline{\boldsymbol{x}}_{i} \bar{y}_{i}\right)$$

• $T\rightarrow \infty$, RE converges to FE
• $Var(\hat{\beta}_{RE})\leq Var(\hat{\beta}_{FE})$

3. First-difference

Another way to eliminate the individual effect.

$$\Delta y_{i t}=\Delta \boldsymbol{x}_{i t}^{\prime} \boldsymbol{\beta}+\Delta \varepsilon_{i t}$$

$$\begin{aligned} \widehat{\boldsymbol{\beta}}_{\Delta} &=\left(\sum_{i=1}^{N} \sum_{t \geq 2} \Delta \boldsymbol{x}_{i t} \Delta \boldsymbol{x}_{i t}^{\prime}\right)^{-1}\left(\sum_{i=1}^{N} \sum_{t \geq 2} \Delta \boldsymbol{x}_{i t} \Delta y_{i t}\right) \\ &=\left(\sum_{i=1}^{N} \Delta \boldsymbol{X}_{i}^{\prime} \Delta \boldsymbol{X}_{i}\right)^{-1}\left(\sum_{i=1}^{N} \Delta \boldsymbol{X}_{i}^{\prime} \Delta \boldsymbol{y}_{i}\right) \\ &=\left(\sum_{i=1}^{N} \boldsymbol{X}_{i}^{\prime} \mathbf{D}_{i}^{\prime} \boldsymbol{D}_{i} \boldsymbol{X}_{i}\right)^{-1}\left(\sum_{i=1}^{N} \boldsymbol{X}_{i}^{\prime} \boldsymbol{D}_{i}^{\prime} \boldsymbol{D}_{i} \boldsymbol{y}_{i}\right) \end{aligned}$$

• T=2, equal to fixed effect estimator
• T>2, not.

4. Dynamic Panel

Weaker identification assumption than previous FE/RE (sequential exogeneity): $x_{it}=y_{it-1}$

$y_{it}=y_{it-1}\beta +\alpha_i + \varepsilon_{it}$ $E(\varepsilon_{it}|x_{it},\cdots,x_{i1}, \alpha_i)=0$

In applications it will often be useful to include time effects $f_t$ to eliminate spurious serial correlation.

Inconsistency FE

The within operator induces correlation between the AR(1) lag and the error. The result is that the within estimator is inconsistent for the coefficients when $T$ is fixed. A thorough explanation appears in Nickell (1981)-incidental parameter problem. $\hat{\beta}_{FE} \xrightarrow{p} \beta + (...)E(x_{it}\varepsilon_{it}-\bar{x}_{i} \varepsilon_{it})=\beta+B^{-1}O(1/\sqrt{T})$

Solutions

• Let $T\rightarrow \infty$
• Anderson-Hsiao Estimator (just-identification)
• Arellano-Bond Estimator (over-identification)

Anderson-Hsiao Estimator

Anderson and Hsiao (1982) made an important breakthrough by showing that a simple instrumental variables estimator is consistent for the parameters.

1. first-differencing to eliminate fixed effects: $\mathbb{E}\left(\Delta y_{i t-1} \Delta \varepsilon_{i t}\right)=\mathbb{E}\left(\left(y_{i t}-y_{i t-1}\right)\left(\varepsilon_{i t}-\varepsilon_{i t-1}\right)\right)=-\sigma_{\varepsilon}^{2}$
2. IV for the endogeniety problem above. Using $y_{it-2}$ for $\Delta y_{it-1}$:
   $$\left(y_{i t-2}, \ldots, y_{i t-p-1}\right) \text { for }\left(\Delta y_{i t-1}, \ldots, \Delta y_{i t-p}\right)$$
3. Given assumption on non-serial correlation of $\varepsilon_{it}$, as $N\rightarrow \infty$,
   $$\widehat{\beta}_{i v} \stackrel{p}{\longrightarrow} \beta-\frac{\mathbb{E}\left(y_{i 1} \Delta \varepsilon_{i 3}\right)}{\mathbb{E}\left(y_{i 1} \Delta y_{i 2}\right)}=\beta$$

Arellano-Bond Estimator

Use $(y_{it-2},y_{it-3},\cdots)$ as IV for $\Delta y_{it-1}$. Apply more valid IV to increase the ==efficiency== (smaller asymptotic variance).

Using these extra instruments has a complication that there are a di§erent number of instruments for each time period. The solution is to view the model as a system of T equations.

Weak IV issue

The Anderson-Hsiao instrument is weak if the $\gamma$ is small Blundell and Bond (1998)

$$\gamma=(\beta-1)\left(\frac{k}{k+\sigma_{\alpha}^{2} / \sigma_{\varepsilon}^{2}}\right), \quad k=\frac{1-\beta}{1+\beta}$$

• unit-root $\beta=1$
• the idiosyncratic effect $\varepsilon$ is small relative to the individual-specific effect $\alpha$

Arellano and Bover (1995) and Blundell and Bond (1998) introduced a set of ==orthogonality conditions== which reduce the weak instrument problem.

5. Probit with fixed effects

$y_{it}^{*} =x_{it}'\beta+\alpha_i+\varepsilon$ log-Likelihood function $l(\beta,\alpha)=\sum_i\sum_t y_{it}log\Phi(x'\beta+\alpha_i)+(1-y_{it})log[1-\Phi(x'\beta+\alpha_i)]$

we cannote difference out the fixed effects from the likelihood function, so estimation on $\beta$ and $\alpha_i$ are required.

2-step

1. $\hat{\alpha}_i(\beta) = \arg\min_{\alpha} l(\beta,\alpha)$
2. $\hat{\beta}=\arg\min_{\beta} l(\beta,\hat{\alpha})$

incidental parameter problem

• $\hat{\alpha}$ only use $T$ obs, non-consistent as $N\rightarrow \infty$.
• the fixed T in $\hat{\alpha}$ contaminate the $\hat{\beta}$ to be inconsistent. $\hat{\beta}_{mle} = \beta + \frac{\beta}{T}+O_p(1/T^2).$
• when $T\rightarrow \infty$, require $\frac{N}{T}\rightarrow \lambda$ that $\sqrt{NT^2}(\hat{\beta}_{mle}-\beta)=\sqrt{\frac{N}{T}}\beta +o_p(1)$

6. Logit with fixed effects

tip

similar to probit case.

7. Multinomial response model

$y_{ij}^{*} =x_{ij}'\beta+z_i\gamma_j+a_{ij}=observed+ \text{choice coefficients} + \text{unobserved factors}$

• non-ordinal choice
• ordinal: bond rating
• Let $V_{ij}=x_{ij}'\beta+z_i\gamma_j$ (mixed logit model)

unordered choice

We model the choice behavior using ==utility maximization== argument. McFadden (1973). $y_i=\arg\max_{j} \{y_{i1}^{*},\cdots,y_{iJ}^{*}\}$ Assume $a_{ij}\sim F(a)=e^{-e^{-a}}$, ==Type 1 extreme value distribution==. $f(a)=e^{-a-e^{-a}}$ Assume $a_{ij}$ independent of $x,z$.

$$\begin{aligned} P(y_i = j|x,z)&=\frac{e^{V_{ij}}}{\sum_{k=1}^J e^{V_{ik}}}\\ &=\frac{e^{x_{ij}\beta+z_i'\gamma_j}}{\sum_{k=1}^J e^{x_{ik}\beta+z_i'\gamma_k}} \end{aligned}$$

Then by MLE.

limitations

• IIA: choice between two is independent of irrelevant alternatives. arise from iid $a_{ij}$ across $1,...,J$. $\frac{P(y_i=j|x)}{P(y_i=k|x)}=e^{(x_{ij}-x_{ik})'\beta}$

8. Interactive effects

$$Y=\sum_{k=1}^{K} \beta_{k}^{0} X_{k}+\varepsilon, \quad \varepsilon=\lambda^{0} f^{0 \prime}+e$$

$$\widehat{\boldsymbol{\beta}}_{R}=\underset{\beta \in \mathbb{R}^{K}}{\operatorname{argmin}} \mathcal{L}_{N T}^{R}(\beta)$$

$$\begin{aligned} \mathcal{L}_{N T}^{R}(\beta) &=\min _{\left\{\Lambda \in \mathbb{R}^{N \times R}, F \in \mathbb{R}^{T \times R}\right\}} \frac{1}{N T}\left\|Y-\beta \cdot X-\Lambda F^{\prime}\right\|_{\mathrm{HS}}^{2} \\ &=\min _{F \in \mathbb{R}^{T \times R}} \frac{1}{N T} \operatorname{Tr}\left[(Y-\beta \cdot X)^{\prime}(Y-\beta \cdot X) M_{F}\right] \\ &=\frac{1}{N T} \sum_{r=R+1}^{T} \mu_{r}\left[(Y-\beta \cdot X)^{\prime}(Y-\beta \cdot X)\right], \end{aligned}$$

The resulting optimization problem for F is a principal components problem, so that the optimal F is given by the R largest principal components. At the optimum, the projector $M_F$ therefore exactly projects out the R largest eigenvalues of this matrix, which gives rise to the final formulation of the profile objective function as the sum over its (T − R) smallest eigenvalues.

strong factor and strong iv

ASSUMPTION SF-Strong Factor Assumption: (i) We have $0<\operatorname{plim}_{N, T \rightarrow \infty} \frac{1}{N} \lambda^{0 \prime} \lambda^{0}<\infty$. (ii) We have $0<\operatorname{plim}_{N, T \rightarrow \infty} \frac{1}{T} f^{0 \prime} f^{0}<\infty$.

9. Suprious Regression

In time-series suprious regression, the suprious regression gives bad result as the estimator is converging to a random variable not a constant.

$$\begin{aligned} &x_{i t}=x_{i, t-1}+\varepsilon_{i t} \\ &y_{i t}=y_{i, t-1}+e_{i t} \end{aligned}$$

In panel model:

$$\begin{gathered} y_{i t}=\alpha+x_{i t} \beta+u_{i t} \\ u_{i t}=\mu_{i}+\nu_{i t} \end{gathered}$$

Kao (1999) shown by sequential asymptotics:

$$\sqrt{n} \widehat{\beta}_{F E} \stackrel{d}{\rightarrow} N\left(0, \frac{2 \sigma_{e}^{2}}{5 \sigma_{\varepsilon}^{2}}\right)$$

In panel, the estimator in the suprious regression gives good inference as it converges to a zero mean distribution.