时间序列复习 chapter 5

密码 2022年6月11日-6月13日

额外的参考资料：北大金融时间序列的备课笔记

Chapter 5 Models with Trend

Deterministic Trend
y t = y 0 + δ t + X t y_t=y_0+\delta t+X_t yt​=y0​+δt+Xt​

Stochastic Trend : Random Walk
Δ y t = ϵ t or  y t = y t − 1 + ϵ t Hence  y t = y 0 + ∑ i = 1 t ϵ i ⏟ stochastic trend  \begin{gathered} \Delta y_{t}=\epsilon_{t} \quad \text { or } \quad y_{t}=y_{t-1}+\epsilon_{t} \\ \text { Hence } y_{t}=y_{0}+\underbrace{\sum_{i=1}^{t} \epsilon_{i}}_{\text {stochastic trend }} \end{gathered} Δyt​=ϵt​ or yt​=yt−1​+ϵt​ Hence yt​=y0​+stochastic trend  i=1∑t​ϵi​​​​

• { ϵ t } \left\{\epsilon_{t}\right\} {ϵt​} is a white noise process.
• y t y_{t} yt​ is said to have a stochastic trend since each ϵ i \epsilon_{i} ϵi​ shock imparts a permanent, albeit random, change in the conditional mean of the series.

Stochastic Trend : Random Walk with Drift
Δ y t = δ + ϵ t or  y t = y t − 1 + δ + ϵ t Hence  y t = y 0 + δ t ⏟ deterministic trend  + ∑ i = 1 t ϵ i ⏟ stochastic trend  \begin{aligned} &\Delta y_{t}=\delta+\epsilon_{t} \text { or } y_{t}=y_{t-1}+\delta+\epsilon_{t} \\ &\text { Hence } y_{t}=y_{0}+\underbrace{\delta t}_{\text {deterministic trend }}+\underbrace{\sum_{i=1}^{t} \epsilon_{i}}_{\text {stochastic trend }} \end{aligned} ​Δyt​=δ+ϵt​ or yt​=yt−1​+δ+ϵt​ Hence yt​=y0​+deterministic trend  δt​​+stochastic trend  i=1∑t​ϵi​​​​

• { ϵ t } \left\{\epsilon_{t}\right\} {ϵt​} is a white noise process.
• obvious tendency to increase or decrease over time

Generalizations of the Stochastic Trend Model
Δ y t = δ + u t or  y t = y t − 1 + δ + u t , \Delta y_{t}=\delta+u_{t} \quad \text { or } y_{t}=y_{t-1}+\delta+u_{t}, Δyt​=δ+ut​ or yt​=yt−1​+δ+ut​,
where u t u_{t} ut​ is a stationary zero mean ARMA ⁡ ( p − 1 , q ) \operatorname{ARMA}(p-1, q) ARMA(p−1,q) process.
Hence,  y t = y 0 + δ t ⏟ deterministic trend  + ∑ i = 1 t u i ⏟ stochastic trend  \text { Hence, } y_{t}=y_{0}+\underbrace{\delta t}_{\text {deterministic trend }}+\underbrace{\sum_{i=1}^{t} u_{i}}_{\text {stochastic trend }}  Hence, yt​=y0​+deterministic trend  δt​​+stochastic trend  i=1∑t​ui​​​

• Eq.(53) includes Eq.(49)( Deterministic Trend )as a special case δ = p − 1 = q = 0 \delta=p-1=q=0 δ=p−1=q=0.
• Eq.(53) includes Eq.(51) as a special case: p − 1 = q = 0 p-1=q=0 p−1=q=0.
• Allow Δ y t \Delta y_{t} Δyt​ to be serially correlated, e.g, logarithm of quarterly GDP (possibly δ > 0 \delta>0 δ>0 ); interest rates (possibly δ = 0 \delta=0 δ=0 ).
• Is X t = ∑ i = 1 t u i X_{t}=\sum_{i=1}^{t} u_{i} Xt​=∑i=1t​ui​ an ARMA process?

Types of Model from the General Formulation y t = α + δ t + X t y_{t}=\alpha+\delta t+X_{t} yt​=α+δt+Xt​, where ϕ ( L ) X t = θ ( L ) ϵ t \phi(L) X_{t}=\theta(L) \epsilon_{t} ϕ(L)Xt​=θ(L)ϵt​

Eq(1): Deterministic Trend

Eq(4): Generalizations of the Stochastic Trend Model

Removing the Deterministic Trend : Detrending

y t = y 0 + δ t + X t y_t=y_0+\delta t+X_t yt​=y0​+δt+Xt​

• 去趋势：在一个常数和时间上回归yt，并保存残差。

• 更一般地说，如果存在多项式趋势（例如二次时间趋势），则通过在确定性多项式时间趋势上回归yt并保存残差来实现去趋势。

• 差分法会丢失一些信息

Removing the Stochastic Trend: Differencing

• Unit root processes (also known as difference stationary processes):
  y t = α + δ t + X t , ϕ ( L ) X t = θ ( L ) ϵ t y_{t}=\alpha+\delta t+X_{t}, \quad \phi(L) X_{t}=\theta(L) \epsilon_{t} yt​=α+δt+Xt​,ϕ(L)Xt​=θ(L)ϵt​
  where ϕ ( L ) = ( 1 − L ) ψ ( L ) \phi(L)=(1-L) \psi(L) ϕ(L)=(1−L)ψ(L), and the roots of ψ ( z ) = 0 \psi(z)=0 ψ(z)=0 lie outside the unit circle.
• We induce stationarity by differencing once :
  ψ ( L ) Δ y t = ψ ( 1 ) δ + θ ( L ) ϵ t . \psi(L) \Delta y_{t}=\psi(1) \delta+\theta(L) \epsilon_{t} . ψ(L)Δyt​=ψ(1)δ+θ(L)ϵt​.
• If we try to detrend a series which has a stochastic trend, then we will not remove the trend.

ARIMA(p,d,q) processes

• We can generalize this concept to consider the case where the series contains more than one “unit root” :
  y t = α + δ t + X t , ( 1 − L ) d ψ ( L ) X t = θ ( L ) ϵ t , y_{t}=\alpha+\delta t+X_{t}, \quad(1-L)^{d} \psi(L) X_{t}=\theta(L) \epsilon_{t}, yt​=α+δt+Xt​,(1−L)dψ(L)Xt​=θ(L)ϵt​,
  where the roots of ψ ( z ) = 0 \psi(z)=0 ψ(z)=0 lie outside the unit circle.
• y t y_{t} yt​ must be differenced d d d times before it becomes stationary
  ψ ( L ) Δ d y t = θ ( L ) ϵ t \psi(L) \Delta^{d} y_{t}=\theta(L) \epsilon_{t} ψ(L)Δdyt​=θ(L)ϵt​
  Then y t y_{t} yt​ is said to be integrated of order d d d. We write y t ∼ I ( d ) y_{t} \sim I(d) yt​∼I(d). So if y t ∼ I ( d ) y_{t} \sim I(d) yt​∼I(d), then △ d y t ∼ I ( 0 ) \triangle^{d} y_{t} \sim I(0) △dyt​∼I(0).
• Taking d d d th differences of an ARIMA(p, d, q) process produces a stationary ARMA ( p , q ) (p, q) (p,q) process.

Test for Unit Root

We consider the following regression model : y t = a 0 + a 1 z t + e t y_{t}=a_{0}+a_{1} z_{t}+e_{t} yt​=a0​+a1​zt​+et​.

• y t , z t ∼ I ( 0 ) y_{t}, z_{t} \sim I(0) yt​,zt​∼I(0)

• y t , z t , e t ∼ I ( 1 ) y_{t}, z_{t}, e_{t} \sim I(1) yt​,zt​,et​∼I(1).

  • This is the case in which the regression is spurious. 残差不平稳

  • In this case, it is often recommended that the regression equation be estimated in first differences.

• y t , z t ∼ I ( 1 ) y_{t}, z_{t} \sim I(1) yt​,zt​∼I(1) and e t ∼ I ( 0 ) e_{t} \sim I(0) et​∼I(0). In this circumstance, y t y_{t} yt​ and z t z_{t} zt​ are cointegrated.

• y t y_{t} yt​ and z t z_{t} zt​ are integrated of different orders. Meaningless regression!

Case 1 : 1: 1: both { y t } \left\{y_{t}\right\} {yt​} and { z t } \left\{z_{t}\right\} {zt​} are stationary

Assumptions of the classical model :

• both { y t } \left\{y_{t}\right\} {yt​} and { z t } \left\{z_{t}\right\} {zt​} sequences are stationary.
• the errors { e t } \left\{e_{t}\right\} {et​} have a zero conditional mean and a finite variance.
• the errors { e t } \left\{e_{t}\right\} {et​} may be serially correlated.

检验有两种方法

1. 中心极限定理：利用之前的AR，MA阶数的test t ( a 0 ^ − 0 ) → N ( 0 , v a r ) \sqrt t(\hat{a_0}-0)\to N(0,var) t ​(a0​^​−0)→N(0,var)
   • 可以使用t检验和F检验
2. simulation 模拟：Monte Carlo

Case 2 : spurious regression

y t = y t − 1 + ϵ y t , ϵ y t ∼ i . i . d . N ( 0 , 1 ) z t = z t − 1 + ϵ z t , ϵ z t ∼ i.i.d.  N ( 0 , 1 ) \begin{array}{ll} y_{t}=y_{t-1}+\epsilon_{y t}, & \epsilon_{y t} \sim \stackrel{i . i . d .}{ } \mathcal{N}(0,1) \\ z_{t}=z_{t-1}+\epsilon_{z t}, & \epsilon_{z t} \sim \text { i.i.d. } \mathcal{N}(0,1) \end{array} yt​=yt−1​+ϵyt​,zt​=zt−1​+ϵzt​,​ϵyt​∼i.i.d.N(0,1)ϵzt​∼ i.i.d. N(0,1)​
where ϵ y t \epsilon_{y t} ϵyt​ and ϵ z t \epsilon_{z t} ϵzt​ are independent, and T = 1000 \mathrm{T}=1000 T=1000.

• Regression result : y ^ t = 23.94 + 0.905 z t \widehat{y}_{t}=23.94+0.905 z_{t} y ​t​=23.94+0.905zt​ with R 2 = 0.4069 R^{2}=0.4069 R2=0.4069. The residuals from the regression are nonstationary.

However,

• since y t y_{t} yt​ and z t z_{t} zt​ are independent, the regression is spurious.

• A spurious regression has a high R 2 R^{2} R2 and t-ratios that appear to be significant, but the results are of no economic meaning.

• The regression output “looks good” because the OLS estimates are inconsistent and the customary tests of statistical inference do not hold.

Case 3 : y t y_t yt​ and z t z_t zt​ are cointegrated

只能用Monte Carlo ，不能用t检验和F检验，因为这个分布函数比较偏不能跟正态分布很好的拟合

两大原因

• spurious
• conditional t test and F test don’t apply

Dickey-Fuller Tests

{ ϵ t } \left\{\epsilon_{t}\right\} {ϵt​} is i.i.d. with mean zero and variance σ 2 \sigma^{2} σ2. Denote ρ ~ = ρ − 1 \tilde{\rho}=\rho-1 ρ~​=ρ−1.（判断是否存在单位根）

• Case 1 : no constant term in the regression
  • y t = ρ y t − 1 + ϵ t y_{t}=\rho y_{t-1}+\epsilon_{t} yt​=ρyt−1​+ϵt​.
  • We test H 0 : ρ = 1 H_{0}: \rho=1 H0​:ρ=1 against H 1 : ρ < 1 H_{1}: \rho<1 H1​:ρ<1.
  • Equivalently, Δ y t = ρ ~ y t − 1 + ϵ t \Delta y_{t}=\tilde{\rho} y_{t-1}+\epsilon_{t} Δyt​=ρ~​yt−1​+ϵt​.
  • We test H 0 : ρ ~ = 0 H_{0}: \tilde{\rho}=0 H0​:ρ~​=0 against H 1 : ρ ~ < 0 H_{1}: \tilde{\rho}<0 H1​:ρ~​<0.
• Case 2 : constant term included in the regression
  • y t = α + ρ y t − 1 + ϵ t y_{t}=\alpha+\rho y_{t-1}+\epsilon_{t} yt​=α+ρyt−1​+ϵt​ (or △ y t = α + ρ ~ y t − 1 + ϵ t \triangle y_{t}=\alpha+\tilde{\rho} y_{t-1}+\epsilon_{t} △yt​=α+ρ~​yt−1​+ϵt​ ).
  • We test H 0 : ρ = 1 H_{0}: \rho=1 H0​:ρ=1 against H 1 : ρ < 1 H_{1}: \rho<1 H1​:ρ<1 (or H 0 : ρ ~ = 0 H_{0}: \tilde{\rho}=0 H0​:ρ~​=0 v.s. H 1 : ρ ~ < 0 H_{1}: \tilde{\rho}<0 H1​:ρ~​<0 ).
• Case 3 : constant term and time trend included in the regression
  • y t = α + β t + ρ y t − 1 + ϵ t y_{t}=\alpha+\beta t+\rho y_{t-1}+\epsilon_{t} yt​=α+βt+ρyt−1​+ϵt​ (or Δ y t = α + β t + ρ ~ y t − 1 + ϵ t ) \left.\Delta y_{t}=\alpha+\beta t+\tilde{\rho} y_{t-1}+\epsilon_{t}\right) Δyt​=α+βt+ρ~​yt−1​+ϵt​).
  • We test H 0 : ρ = 1 H_{0}: \rho=1 H0​:ρ=1 against H 1 : ρ < 1 H_{1}: \rho<1 H1​:ρ<1 (or H 0 : ρ ~ = 0 H_{0}: \tilde{\rho}=0 H0​:ρ~​=0 v.s. H 1 : ρ ~ < 0 ) \left.H_{1}: \tilde{\rho}<0\right) H1​:ρ~​<0).

上述回归中的两个估计量都没有极限正态分布。case2,3 也不能使用通常的F检验

Analysis of Case 1

• H 0 : y t = y t − 1 + ϵ t H_{0}: y_{t}=y_{t-1}+\epsilon_{t} H0​:yt​=yt−1​+ϵt​.

• No constant term in the test regression : y t = ρ y t − 1 + ϵ t y_{t}=\rho y_{t-1}+\epsilon_{t} yt​=ρyt−1​+ϵt​. Then the OLS estimator ρ T ^ = ∑ t = 1 T y t − 1 y t ∑ t = 1 T y t − 1 2 \hat{\rho_{T}}=\frac{\sum_{t=1}^{T} y_{t-1} y_{t}}{\sum_{t=1}^{T} y_{t-1}^{2}} ρT​^​=∑t=1T​yt−12​∑t=1T​yt−1​yt​​.

• To get better sense of the asymptotic of ρ T ^ \hat{\rho_{T}} ρT​^​, we derive
  ρ T ^ − 1 = ∑ t = 1 T y t − 1 ϵ t ∑ t = 1 T y t − 1 2 . \hat{\rho_{T}}-1=\frac{\sum_{t=1}^{T} y_{t-1} \epsilon_{t}}{\sum_{t=1}^{T} y_{t-1}^{2}} . ρT​^​−1=∑t=1T​yt−12​∑t=1T​yt−1​ϵt​​.
  The denominator is O p ( T 2 ) O_{p}\left(T^{2}\right) Op​(T2), while the numerator is O p ( T ) O_{p}(T) Op​(T) ( F C L T ) (F C L T) (FCLT)

• So it makes sense to multiply this equation by T T T
  T ( ρ T ^ − 1 ) = 1 T ∑ t = 1 T y t − 1 ϵ t 1 T 2 ∑ t = 1 T y t − 1 2 . T\left(\hat{\rho_{T}}-1\right)=\frac{\frac{1}{T} \sum_{t=1}^{T} y_{t-1} \epsilon_{t}}{\frac{1}{T^{2}} \sum_{t=1}^{T} y_{t-1}^{2}} . T(ρT​^​−1)=T21​∑t=1T​yt−12​T1​∑t=1T​yt−1​ϵt​​.

• By FCLT and continuous mapping theorem,
  T ( ρ T ^ − 1 ) ⟶ D 1 2 ( W 2 ( 1 ) − 1 ) ∫ 0 1 W 2 ( r ) d r superconsistent.  T\left(\hat{\rho_{T}}-1\right) \stackrel{\mathcal{D}}{\longrightarrow} \frac{\frac{1}{2}\left(W^{2}(1)-1\right)}{\int_{0}^{1} W^{2}(r) d r} \text { superconsistent. } T(ρT​^​−1)⟶D​∫01​W2(r)dr21​(W2(1)−1)​ superconsistent.

• The probability that T ( ρ T ^ − 1 ) T\left(\hat{\rho_{T}}-1\right) T(ρT​^​−1) is negative approaches 0.68 0.68 0.68 as T T T becomes large. The limiting distribution of T ( ρ T ^ − 1 ) T\left(\hat{\rho_{T}}-1\right) T(ρT​^​−1) is skewed to the left.

• OLS t t t test of H 0 : ρ = 1 H_{0}: \rho=1 H0​:ρ=1 :
  t T = ρ T ^ − 1 s T / ∑ t = 1 T y t − 1 2 D i c k e y F u l l e r t e s t t_{T}=\frac{\hat{\rho_{T}}-1}{s_{T} / \sqrt{\sum_{t=1}^{T} y_{t-1}^{2}}}\quad Dickey ~~Fuller~~ test tT​=sT​/∑t=1T​yt−12​ ​ρT​^​−1​Dickey  Fuller  test
  where s T / ∑ t = 1 T y t − 1 2 s_{T} / \sqrt{\sum_{t=1}^{T} y_{t-1}^{2}} sT​/∑t=1T​yt−12​ ​ is the usual OLS standard error for the estimated coefficient, and s T 2 = ∑ t = 1 T ( y t − ρ T ^ y t − 1 ) 2 / ( T − 1 ) s_{T}^{2}=\sum_{t=1}^{T}\left(y_{t}-\hat{\rho_{T}} y_{t-1}\right)^{2} /(T-1) sT2​=∑t=1T​(yt​−ρT​^​yt−1​)2/(T−1) is the OLS estimate of the residual variance.

• Again, we can use FCLT and continuous mapping theorem to show

t T ⟶ D 1 2 ( W 2 ( 1 ) − 1 ) [ ∫ 0 1 W 2 ( r ) d r ] 1 / 2 . t_{T} \stackrel{\mathcal{D}}{\longrightarrow} \frac{\frac{1}{2}\left(W^{2}(1)-1\right)}{\left[\int_{0}^{1} W^{2}(r) d r\right]^{1 / 2}} . tT​⟶D​[∫01​W2(r)dr]1/221​(W2(1)−1)​.

Augmented Dickey-Fuller Tests for Unit Roots

ADF检验的基本思想是通过在回归中包含高阶自回归项来控制序列相关性。

Case I: no constant term in the regression.

We can rewrite the model as
y t = ρ y t − 1 + η 1 Δ y t − 1 + ⋯ + η p − 1 Δ y t − p + 1 + ϵ t , or  Δ y t = ρ ~ y t − 1 + η 1 Δ y t − 1 + ⋯ + η p − 1 Δ y t − p + 1 + ϵ t \begin{aligned} y_{t} &=\rho y_{t-1}+\eta_{1} \Delta y_{t-1}+\cdots+\eta_{p-1} \Delta y_{t-p+1}+\epsilon_{t}, \\ \text { or } \Delta y_{t} &=\tilde{\rho} y_{t-1}+\eta_{1} \Delta y_{t-1}+\cdots+\eta_{p-1} \Delta y_{t-p+1}+\epsilon_{t} \end{aligned} yt​ or Δyt​​=ρyt−1​+η1​Δyt−1​+⋯+ηp−1​Δyt−p+1​+ϵt​,=ρ~​yt−1​+η1​Δyt−1​+⋯+ηp−1​Δyt−p+1​+ϵt​​
where ρ ≡ ϕ 1 + ϕ 2 + ⋯ + ϕ p , η j ≡ − [ ϕ j + 1 + ϕ j + 2 + ⋯ + ϕ p ] \rho \equiv \phi_{1}+\phi_{2}+\cdots+\phi_{p}, \eta_{j} \equiv-\left[\phi_{j+1}+\phi_{j+2}+\cdots+\phi_{p}\right] ρ≡ϕ1​+ϕ2​+⋯+ϕp​,ηj​≡−[ϕj+1​+ϕj+2​+⋯+ϕp​] for j = 1 , 2 , ⋯ , p − 1 j=1,2, \cdots, p-1 j=1,2,⋯,p−1, and ρ ~ ≡ ρ − 1 \tilde{\rho} \equiv \rho-1 ρ~​≡ρ−1.

• It can be shown that testing unit roots is equivalent to testing ρ = 1 ( \rho=1( ρ=1( or ρ ~ = 0 ) \tilde{\rho}=0) ρ~​=0).
• We test H 0 : ρ = 1 H_{0}: \rho=1 H0​:ρ=1 against H 1 : ρ < 1 H_{1}: \rho<1 H1​:ρ<1 (or H 0 : ρ ~ = 0 H_{0}: \tilde{\rho}=0 H0​:ρ~​=0 v.s. H 1 : ρ ~ < 0 ) \left.H_{1}: \tilde{\rho}<0\right) H1​:ρ~​<0)

Case II : constant term included in the regression

y t = α + ρ y t − 1 + η 1 Δ y t − 1 + ⋯ + η p − 1 △ y t − p + 1 + ϵ t y_{t}=\alpha+\rho y_{t-1}+\eta_{1} \Delta y_{t-1}+\cdots+\eta_{p-1} \triangle y_{t-p+1}+\epsilon_{t} yt​=α+ρyt−1​+η1​Δyt−1​+⋯+ηp−1​△yt−p+1​+ϵt​ (or Δ y t = α + ρ ~ y t − 1 + η 1 Δ y t − 1 + ⋯ + η p − 1 Δ y t − p + 1 + ϵ t \Delta y_{t}=\alpha+\tilde{\rho} y_{t-1}+\eta_{1} \Delta y_{t-1}+\cdots+\eta_{p-1} \Delta y_{t-p+1}+\epsilon_{t} Δyt​=α+ρ~​yt−1​+η1​Δyt−1​+⋯+ηp−1​Δyt−p+1​+ϵt​.

We test H 0 : ρ = 1 H_{0}: \rho=1 H0​:ρ=1 against H 1 : ρ < 1 H_{1}: \rho<1 H1​:ρ<1 (or H 0 : ρ ~ = 0 H_{0}: \tilde{\rho}=0 H0​:ρ~​=0 v.s. H 1 : ρ ~ < 0 ) \left.H_{1}: \tilde{\rho}<0\right) H1​:ρ~​<0)

Case III: constant term and time trend included in the regression
y t = α + β t + ρ y t − 1 + η 1 Δ y t − 1 + ⋯ + η p − 1 Δ y t − p + 1 + ϵ t y_{t}=\alpha+\beta t+\rho y_{t-1}+\eta_{1} \Delta y_{t-1}+\cdots+\eta_{p-1} \Delta y_{t-p+1}+\epsilon_{t} yt​=α+βt+ρyt−1​+η1​Δyt−1​+⋯+ηp−1​Δyt−p+1​+ϵt​ (or Δ y t = α + β t + ρ ~ y t − 1 + η 1 Δ y t − 1 + ⋯ + η p − 1 Δ y t − p + 1 + ϵ t \Delta y_{t}=\alpha+\beta t+\tilde{\rho} y_{t-1}+\eta_{1} \Delta y_{t-1}+\cdots+\eta_{p-1} \Delta y_{t-p+1}+\epsilon_{t} Δyt​=α+βt+ρ~​yt−1​+η1​Δyt−1​+⋯+ηp−1​Δyt−p+1​+ϵt​.

We test H 0 : ρ = 1 H_{0}: \rho=1 H0​:ρ=1 against H 1 : ρ < 1 H_{1}: \rho<1 H1​:ρ<1 (or H 0 : ρ ~ = 0 H_{0}: \tilde{\rho}=0 H0​:ρ~​=0 v.s. H 1 : ρ ~ < 0 ) \left.H_{1}: \tilde{\rho}<0\right) H1​:ρ~​<0)

以上t检验和F检验不需要为序列相关进行调整，直接用通常的t检验和f检验即可

Testing for Higher Orders of Integration

Other Approaches to Testing for Unit Roots

• PP test - Phillips, P.C.B. and Perron, P. (1988). “Testing for a Unit Root in Time Series Regression.”
• DF-GLS test - Elliott, G., Rothenberg, T.J. and Stock, J.H. (1996). “Efficient Tests for an Autoregressive Unit Root.”
• Hylleberg, S., Engle, R.F., Granger, C.W.J. and Yoo, B.S. ( 1990 ) (1990) (1990). “Seasonal Integration and Cointegration.”
• Panel unit root test-Im, K.S., Pesaran, M.H., and Shin, Y. (2003), “Testing for Unit Roots in Heterogeneous Panels.”

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