Consistent Model Selection: Over Rolling Windows

Abstract

In this chapter we analyze asymptotic properties of the simulated out-of-sample predictive mean squared error (PMSE) criterion based on a rolling window when selecting among nested forecasting models. When the window size is a fixed fraction of the sample size, Inoue and Kilian (J Econ 130: 273–306, 2006) show that the PMSE criterion is inconsistent. We consider alternative schemes under which the rolling PMSE criterion is consistent. When the window size diverges slower than the sample size at a suitable rate, we show that the rolling PMSE criterion selects the correct model with probability approaching one when parameters are constant or when they are time varying. We provide Monte Carlo evidence and illustrate the usefulness of the proposed methods in forecasting inflation.

Appendix

1.1 A.1 Lemmas

Next, we present a lemma similar to Lemma A2 of Clark and McCracken (2000).

Lemma 1

Suppose that Assumptions 1 and 2 hold and that $\gamma =0$. Then:

• (a) $\frac{1}{T-h-W} \sum _{t=W+1}^{T-h}u_{t+h}x_{t}B_{1}(t)H_{1}(t) =o_{p}\left(\frac{1}{W} \right)$.

• (b) $\frac{1}{T-h-W} \sum _{t=W+1}^{T-h}v_{t+h}z_{t}B_{2}(t)H_{2}(t) =o_{p}\left(\frac{1}{W} \right)$.

• (c) $\frac{1}{T-h-W}\sum \limits _{t=W+1}^{T-h}H_1^{\prime }(t)B_1(t) x_tx_t^{\prime }B_1(t)H_1(t)=\frac{1}{T-h-W}\sum \limits _{t=W+h}^{T-1}H_1^{\prime }(t)B_1H_1(t)+o_p\left( \frac{1}{W}\right)$.

• (d) $\frac{1}{T-h-W}\sum \limits _{t=W+1}^{T-h}H_2^{\prime }(t)B_2(t) z_tz_t^{\prime }B_2(t)H_2(t) = \frac{1}{T-h-W}\sum \limits _{t=W+1}^{T-h}H_2^{\prime }(t)B_2H_2(t)+o_{p}\left( \frac{1}{W}\right)$.

Proof of Lemma 1: The proofs for (a) and (c) are very similar to those for (b) and (d), respectively. For brevity, we only provide the proofs of (b) and (d). The results for (a) and (c) can be easily derived by replacing $z_{t}$ and $\beta$ by $x_{t}$ and $\alpha$, respectively.

Note that

$$\begin{aligned} \frac{1}{T-h-W}\sum _{t=W+1}^{T-h}v_{t+h}z_{t}B_{2}(t)H_{2}(t)&= \frac{1}{T-h-W}\sum _{t=W+1}^{T-h}v_{t+h}z_{t}B_{2}H_{2}(t) \\&\quad +\frac{1}{T-h-W}\sum _{t=W+1}^{T-h}v_{t+h}z_{t}(B_{2}(t)-B_{2})H_{2}(t) \end{aligned}$$

By Assumption 2(b) and Hölder’s inequality, the second moments of the summands on the right-hand side are of order $O(W^{-1})$ and $O(W^{-2})$, respectively. Thus, it follows from Assumption 2(c) that the variance of the left-hand side is of order $O(T^{-1}W^{-1})$. By the Chebyshev inequality and Assumption 1, the left-hand side is $o_{p}(W^{-1})$.

The proof of (d) is composed of two stages. In the first stage, we show that $B_2(t)$ in the equation can be approximated by its expectation $B_2$, which is

$$\begin{aligned}&\frac{1}{T-h-W}\sum \limits _{t=W+1}^{T-h}H_2^{\prime }(t)B_2(t) z_tz_t^{\prime }B_{2}(t)H_2(t) \nonumber \\&\quad =\frac{1}{T-h-W}\sum \limits _{t=W+1}^{T-h}H_2^{\prime }(t)B_{2} z_tz_t^{\prime }B_{2}H_2(t)+o_p\left(\frac{1}{W}\right) \end{aligned}$$

(20)

Since the left-hand side of Eq. (A.1) contains four terms,

$$\begin{aligned}&\frac{1}{T-h-W} \sum \limits _{t=W+1}^{T-h}H_2^{\prime }(t)B_2(t) z_tz_t^{\prime }B_2(t)H_2(t) \nonumber \\&\quad \qquad = \frac{1}{T-h-W} \sum \limits _{t=W+1}^{T-h}H_2^{\prime }(t)B_2z_tz_t^{\prime }B_2H_2(t) \nonumber \nonumber \\&\quad \qquad \quad \quad + \frac{1}{T-h-W} \sum \limits _{t=W+1}^{T-h}H_2^{\prime }(t)(B_2(t)-B_2)z_tz_t^{\prime }(B_2(t)-B_2)H_2(t) \nonumber \nonumber \\&\quad \qquad \quad \quad + \frac{1}{T-h-W} \sum \limits _{t=W+1}^{T-h}H_2^{\prime }(t)B_2z_tz_t^{\prime }(B_2(t)-B_2)H_2(t) \nonumber \nonumber \\&\quad \qquad \quad \quad + \frac{1}{T-h-W} \sum \limits _{t=W+1}^{T-h}H_2^{\prime }(t)(B_2(t)-B_2)z_tz_t^{\prime }B_2H_2(t), \end{aligned}$$

(21)

which include the first term in the right-hand side of Eq.  (A.1).

By Assumption 2(b) and Hölder’s inequality, the second moments of the summands in the last three terms are of order $O(W^{-4})$, $O(W^{-3})$, and $O(W^{-3})$, respectively. Thus, their first moments are at most $O(W^{-3})=o(W^{-1})$. By using these and Assumption 2(e), the second moments of the last three terms are thus of the order $O(T^{-1}W^{-4})$, $O(T^{-1}W^{-3})$ and $O(T^{-1}W^{-1})$, respectively. By the Chebyshev inequality and Assumption 1, these last three terms are of the order $o_{p}(W^{-1})$, proving (A.1).

The second stage of the proof of (d) is to show that we can further approximate $z_tz_t^{\prime }$ in the first term in the right-hand side of Eq. (A.2) by its expectation $E(z_tz_t^{\prime })$. Adding and subtracting $E(z_t z_t^{\prime })$, we obtain

$$\begin{aligned}&\frac{1}{T-h-W} \sum \limits _{t=W+1}^{T-h}H_2^{\prime }(t)B_2z_tz_t^{\prime }B_2H_2(t)\nonumber \\&\qquad =\frac{1}{T-h-W} \sum \limits _{t=W+1}^{T-h}H_2^{\prime }(t)B_2E(z_tz_t^{\prime })B_2H_2(t) \nonumber \nonumber \\&\qquad \quad \quad + \frac{1}{T-h-W} \sum \limits _{t=W+1}^{T-h}H_2^{\prime }(t)B_2(z_tz_t^{\prime }-E(z_tz_t^{\prime }))B_2 H_2(t) \end{aligned}$$

(22)

The mean of the second term is $o_{p}(W^{-1})$ by Assumption 2(d). The second moments of the summand in the second term is $O(W^{-2})$ by Assumption 2(b). Using these and Assumption 2(e), the second moment of the second term is of the order $o(W^{-2})$. By the Chebyshev inequality, () is $o_{p}(W^{-1})$.

Lemma 2.

Suppose that Assumptions 3 and 4 hold and that $\gamma (\cdot )=0$.

• (a) $\frac{1}{T-h-W} \sum _{t=W+1}^{T-h}u_{T,t+h}x_{T,t}B_{1}(t)H_{1}(t) =o_{p}\left(\frac{1}{W} \right)$.

• (b) $\frac{1}{T-h-W} \sum _{t=W+1}^{T-h}v_{T,t+h}z_{T,t}B_{2}(t)H_{2}(t) =o_{p}\left(\frac{1}{W} \right)$.

• (c) $\frac{1}{T-h-W}\sum \limits _{t=W+1}^{T-h}H_1^{\prime }(t)B_1(t) x_{T,t}x_{T,t}^{\prime }B_1(t)H_1(t) = \frac{1}{T-h-W}\sum \limits _{t=W+1}^{T-h}H_1^{\prime }(t)\bar{ B}_1\left(\frac{t}{T}\right)H_1(t)+o_p\left(\frac{1}{W}\right)$.

• (d) $\frac{1}{T-h-W}\sum \limits _{t=W+1}^{T-h}H_2^{\prime }(t)B_2(t) z_{T,t}z_{T,t}^{\prime }B_2(t)H_2(t) = \frac{1}{T-h-W}\sum \limits _{t=W+1}^{T-h}H_2^{\prime }(t)\bar{B}_2\left(\frac{t}{T}\right)H_2(t)+o_{p}\left(\frac{1}{W}\right)$.

Proof of Lemma 2 Under Assumptions 3 and 4 the proof of Lemma 2 takes exactly the same steps as the proof of Lemma 1 except that $B_{i}$, $u_{t}$, and $v_{t}$ are replaced by $\bar{B}_{i}\left(\frac{t}{T}\right)$, $u_{T,t}$, and $v_{T,t}$, respectively. This is because Lemma 2 is written in terms of $u_{T,t}$ and $v_{T,t}$ rather than in terms of $\hat{\alpha }_{t,W}-\alpha \left(\frac{t}{T} \right)$ and $\hat{\beta }_{t,W}-\beta \left(\frac{t}{T}\right)$ which we deal with in the proof of Theorem 2.

1.2 A.2 Proofs of Theorems

Proof of Theorem 1 Note that the PMSEs $\hat{\sigma }_{1,W}^{2}$ and $\hat{\sigma }_{2,W}^{2}$ can be expanded as

$$\begin{aligned} \hat{\sigma }_{1,W}^{2}&=\frac{1}{T-h-W}\sum \limits _{t=W+1}^{T-h}{\left( { y_{t+h}-\widehat{\alpha }_{t}^{\prime }x_{t}}\right) }^{2} \nonumber \\&=\frac{1}{T-h-W}\sum \limits _{t=W+1}^{T-h}{\left( y_{t+h}-{\alpha ^{*}} ^{\prime }x_{t}-\left( \widehat{\alpha }_{t}^{\prime }x_{t}-{\alpha ^{*}}^{\prime }x_{t}\right) \right) ^{2}} \nonumber \\&=\frac{1}{T-h-W}\sum \limits _{t=W+1}^{T-h}{\left( y_{t+h}-{\alpha ^{*}} ^{\prime }x_{t}\right) ^{2}} \nonumber \\&\qquad -\frac{2}{T-h-W}\sum \limits _{t=W+1}^{T-h}{ \left( y_{t+h}-{\alpha ^{*}}^{\prime }x_{t}\right) x_{t}^{\prime }\left( \widehat{\alpha }_{t}-{\alpha ^{*}}\right) } \nonumber \\&\qquad +\frac{1}{T-h-W}\sum \limits _{t=W+1}^{T-h}{\left( \widehat{\alpha } _{t}^{\prime }-{\alpha ^{*}}^{\prime }\right) x_{t}x_{t}^{\prime }\left( \widehat{\alpha }_{t}-{\alpha ^{*}}\right) } \end{aligned}$$

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and

$$\begin{aligned} \hat{\sigma }_{2,W}^{2}&=\frac{1}{T-h-W} \sum \limits _{t=W+1}^{T-h}{\left( { y_{t+h}-\widehat{\beta }_{t}^{\prime }z_{t}}\right) }^{2} \nonumber \\&=\frac{1}{T-h-W}\sum \limits _{t=W+1}^{T-h}{\left( {y_{t+h}-\beta ^{\prime }z_{t}-\left( {\widehat{\beta }_{t}^{\prime }z_{t}-\beta ^{\prime }z_{t}} \right) }\right) }^{2} \nonumber \\&=\frac{1}{T-h-W}\sum \limits _{t=W+1}^{T-h}{\left( {y_{t+h}-\beta ^{\prime }z_{t}}\right) }^{2} \nonumber \\&\qquad -\frac{2}{T-h-W} \sum \limits _{t=W+1}^{T-h}{\left( { y_{t+h}-\beta ^{\prime }z_{t}}\right) }z_{t}^{\prime }\left( {\widehat{\beta }_{t}-\beta }\right) \nonumber \\&\qquad +\frac{1}{T-h-W}\sum \limits _{t=W+1}^{T-h}{\left( {\widehat{\beta } _{t}^{\prime }-\beta ^{\prime }}\right) }z_{t}z_{t}^{\prime }\left( { \widehat{\beta }_{t}-\beta }\right) , \end{aligned}$$

(24)

respectively, where $\alpha ^{*}=[E(x_{t}x_{t}^{\prime })]^{-1}E(x_{t}y_{t+h})$. There are two cases: the case in which the data are generated from model 1, i.e., $\gamma =0$ (case 1) and the case in which the data are generated from model 2, i.e., $\gamma \ne 0$ (case 2).

In case 1, the actual model is $y_{t+h}=\alpha ^{\prime }x_{t}+v_{t+h}$. The first component of $\hat{\sigma }_{2,W}^{2}$ in Eq.  (A.5) is numerically identical to the first component of $\hat{\sigma }_{1,W}^{2}$ in Eq.  (A.4) because $\gamma =0$ and $\alpha -\alpha ^{*}=0$. Note that all the other components converge to zero faster since all parameters are consistently estimated. Under the case where Model 1 is true, the difference between the probability limit of $\hat{\sigma }_{1,W}^{2}$ and $\hat{\sigma }_{2,W}^{2}$ is zero, which does not identify which model is the true model. Only comparing the probability limits of $\hat{\sigma }_{1,W}^{2}$ and $\hat{\sigma }_{2,W}^{2}$ as $T$ and $W$ go to infinity and $W$ diverges slowly than $T$ is not sufficient for the model selection to indicate that $\lim _{T\rightarrow \infty ,\ W\rightarrow \infty }P(\hat{\sigma }_{1,W}^{2}<\hat{\sigma } _{2,W}^{2})=1$. However, if we can tell whether $\hat{\sigma }_{1,W}^{2}$ is always smaller than $\hat{\sigma }_{2,W}^{2}$ along the path of convergence of $T$ and $W$ toward infinity, the true model can still be identified. Since the models are nested $u_{t+h}=v_{t+h}$, it follows from () and (A.5) that

$$\begin{aligned} \hat{\sigma }_{2,W}^{2}-\hat{\sigma }_{1,W}^{2}&=\frac{2}{T-h-W} \sum _{t=W+1}^{T-h}\left[ v_{t+h}z_{t}^{\prime }(\widehat{\beta }_{t}-\beta )-v_{t+h}x_{t}^{\prime }(\widehat{\alpha }_{t}-\alpha )\right] \nonumber \\&\quad +\frac{1}{T-h-W}\sum \limits _{t=W+1}^{T-h} \left[ {\left( {\widehat{\beta } _{t}^{\prime }-\beta ^{\prime }}\right) }z_{t}z_{t}^{\prime }\left( { \widehat{\beta }_{t}-\beta }\right) \right. \nonumber \\&\qquad \qquad \qquad \qquad \qquad \qquad \left. -{\left( {\widehat{\alpha }_{t}^{\prime }-{\alpha }^{\prime }}\right) }x_{t}x_{t}^{\prime }\left( {\widehat{\alpha } _{t}-{\alpha }}\right) \right] \nonumber \\&=\frac{2}{T-h-W}\sum _{t=W+1}^{T-h}\left[ v_{t+h}z_{t}^{\prime }B_{2}(t)H_{2}(t)-v_{t+h}x_{t}^{\prime }B_{1}(t)H_{1}(t)\right] \nonumber \\&\quad +\frac{1}{T-h-W}\sum _{t=W+1}^{T-h}\left[ H_{2}(t)^{\prime }B_{2}(t)z_{t}z_{t}^{\prime }B_{2}(t)H_{2}(t)\right. \nonumber \\&\qquad \qquad \qquad \qquad \qquad \qquad \left.-H_{1}(t)^{\prime }B_{1}(t)x_{t}x_{t}^{\prime }B_{1}(t)H_{1}(t)\right] \nonumber \\&=\frac{1}{T-h-W}\sum _{t=W+1}^{T-h}\left[ H_{2}(t)^{\prime }B_{2}H_{2}(t)-H_{1}(t)^{\prime }B_{1}H_{1}(t)\right] +o_{p}\left( \frac{1}{W }\right) \end{aligned}$$

(25)

where the last equality follows from Lemma 1(a)–(d).

To get the sign of Eq. (A.6), we first define $Q$ by

$$\begin{aligned} Q\;=\;[E(z_{t}z_{t}^{\prime })]^{\frac{1}{2}}\left\{ [E(z_{t}z_{t}^{\prime })]^{-1}-\left[ \begin{array}{cc} [E(x_{t}x_{t}^{\prime })]^{-1}&\mathbf 0 _{l\times (k-l)} \\ \mathbf 0 _{(k-l)\times l}&\mathbf 0 _{(k-l)\times (k-l)} \end{array} \right] \right\} [E(z_{t}z_{t}^{\prime })]^{\frac{1}{2}} \end{aligned}$$

(26)

as in Lemma A.4 of Clark and McCracken (2000). Clark and McCracken (2000) show that the $Q$ matrix is symmetric and idempotent. An idempotent matrix is positive semidefinite, which means for all $v\in \mathfrak R ^{k}$, $v^{T}Qv\ge 0$. It implies that

$$\begin{aligned}&\quad \left[ \frac{1}{W_{h}^{\frac{1}{2}}}\sum \limits _{s=t-W}^{t-h}z_{s}v_{s+h} \right] ^{\prime }[E(z_{t}z_{t}^{\prime })]^{-1}\left[ \frac{1}{W_{h}^{\frac{ 1}{2}}}\sum \limits _{s=t-W}^{t-h}z_{s}v_{s+h}\right] \nonumber \\&\quad -\left[ \frac{1}{W_{h}^{\frac{1}{2}}}\sum \limits _{s=t-W}^{t-h}x_{s}v_{s+h} \right] ^{\prime }[E(x_{t}x_{t}^{\prime })]^{-1}\left[ \frac{1}{W_{h}^{\frac{ 1}{2}}}\sum \limits _{s=t-W}^{t-h}x_{s}v_{s+h}\right] \nonumber \\&=\left[ \frac{1}{W_{h}^{\frac{1}{2}}}\sum \limits _{s=t-W}^{t-h}z_{s}v_{s+h} \right] ^{\prime }\left\{ [E(z_{t}z_{t}^{\prime })]^{-1}-\left[ \begin{array}{cc} [E(x_{t}x_{t}^{\prime })]^{-1}&\mathbf 0 _{l\times (k-l)} \\ \mathbf 0 _{(k-l)\times l}&\mathbf 0 _{(k-l)\times (k-l)} \end{array} \right] \right\} \nonumber \\&\qquad \times \left[ \frac{1}{W_{h}^{\frac{1}{2}}}\sum \limits _{s=t-W}^{t-h}z_{s}v_{s+h}\right] \nonumber \\&=\left[ \frac{1}{W_{h}^{\frac{1}{2}}}\sum \limits _{s=t-W}^{t-h}z_{s}v_{s+h} \right] ^{\prime }[E(z_{t}z_{t}^{\prime })]^{-\frac{1}{2}}\cdot Q\cdot [E(z_{t}z_{t}^{\prime })]^{-\frac{1}{2}} \nonumber \\&\qquad \times \left[ \frac{1}{W_{h}^{\frac{1}{2}}}\sum \limits _{s=t-W}^{t-h}z_{s}v_{s+h}\right]\;\;\ge \;\;0 \end{aligned}$$

(27)

Note that the probability that $[E(z_{t}z_{t}^{\prime })]^{-1/2}W_{h}^{-1/2}\sum _{s=t-W}^{t-h}z_{s}v_{s+h}$ lies in the null space of $Q$ for infinitely many $t$ approaches zero because the dimension of the null space is $l<k$. Thus, the average of (A.8) over $t$ is positive with probability approaching one. Combining the results in Eqs.  (A.6) and (A.8), we find that the sign of $W(\hat{\sigma }_{2,W}^{2}-\hat{\sigma }_{1,W}^{2})$ is always positive with probability approaching one. Therefore, when $\gamma =0$, $\hat{\sigma } _{1,W}^{2}<\hat{\sigma }_{2,W}^{2}$ with probability approaching one.

In case 2, that is, when Model 2 is the true model, we have $y_{t+h}$ $=\beta ^{\prime }z_{t}+v_{t+h}$ $=\alpha ^{\prime }x_{t}+\gamma ^{\prime }w_{t}+v_{t+h}$. By Assumptions 2(a)(b), the second and third terms on the right-hand sides of () and (A.5) are both $o_{p}(T^{1/2}/W)$ and $o_{p}(T/W^{2})$, respectively. Thus, they are $o_{p}(1)$ by Assumption 1. The first term on the right-hand side of Eq. (A.5) converges to the variance of $v_{t+h}$ as the sample size $T$ goes to infinity:

$$\begin{aligned} \frac{1}{T-h-W}\sum \limits _{t=W+1}^{T-h}{\left( {y_{t+h}-\beta ^{\prime }z_{t}}\right) }^{2}=\frac{1}{T-h-W}\sum \limits _{t=W+1}^{T-h}{v_{t+h}^{2}} \overset{p}{\rightarrow }\sigma _{2}^{2}. \end{aligned}$$

(28)

Similarly, the first term on the right-hand side of Eq. () converges in probability to the variance of $u_{t+h}\equiv y_{t+h}-\alpha ^{*\prime }x_{t}$:

$$\begin{aligned} \hat{\sigma }_{1,W}^{2}&=\frac{1}{T-h-W}\sum \limits _{t=W+h}^{T-1}{\left( y_{t+h}-{\alpha ^{*}}^{\prime }x_{t}\right) ^{2}}+o_{p}(1) \nonumber \\&\overset{p}{\rightarrow } E\left[ (y_{t+h}-{\alpha ^{*}}^{\prime }x_{t})^{2}\right] \nonumber \\&=E\left[ (\alpha ^{\prime }x_{t}+\gamma ^{\prime }w_{t}+v_{t+h}-{\alpha ^{*}}^{\prime }x_{t})^{2}\right] \nonumber \\&=E\left[ (v_{t+h}+(\alpha ^{\prime }-{\alpha ^{*}}^{\prime })x_{t}+\gamma ^{\prime }w_{t})^{2}\right] \nonumber \\&=\sigma _{2}^{2}+\left[ \begin{array}{c} \alpha -\alpha ^{*} \\ \gamma \end{array} \right] ^{\prime }\left[ \begin{array}{cc} E(x_{t}x_{t}^{\prime })&E(x_{t}w_{t}^{\prime }) \\ E(w_{t}x_{t}^{\prime })&E(w_{t}w_{t}^{\prime }) \end{array} \right] \left[ \begin{array}{c} \alpha -\alpha ^{*} \\ \gamma \end{array} \right]\;\;>\;\;\sigma _{2}^{2}. \end{aligned}$$

(29)

Therefore, when Model 2 is true, the PMSEs satisfy $P(\hat{\sigma }_{1,W}^{2}> \hat{\sigma }_{2,W}^{2})=1$ as $T\rightarrow \infty$ and $W\rightarrow \infty$, where $W$ diverges slower than $T$.

Proof of Theorem 2 Note that the PMSEs, $\hat{\sigma }_{1,W}^{2}$ and $\hat{\sigma }_{2,W}^{2}$ can be expanded as

$$\begin{aligned} \hat{\sigma }_{1,W}^{2}&=\frac{1}{T-h-W}\sum \limits _{t=W+1}^{T-h}\left( y_{T,t+h}-\alpha ^{*}\left(\frac{t}{T}\right) ^{\prime }x_{T,t}\right)^{2} \nonumber \\&\quad -\frac{2}{T-h-W}\sum \limits _{t=W+1}^{T-h}\left( y_{T,t+h}-\alpha ^{*}\left(\frac{t}{T}\right)^{\prime }x_{T,t}\right) x_{T,t}^{\prime } \left( \widehat{\alpha }_{t}-\alpha ^{*}\left(\frac{t}{T}\right)\right)\nonumber \\&\quad +\frac{1}{T-h-W}\sum \limits _{t=W+1}^{T-h}\left(\widehat{\alpha } _{t}^{\prime }-\alpha ^{*}\left(\frac{t}{T}\right)\right)^{\prime } x_{T,t}x_{T,t}^{\prime }\left( \widehat{\alpha }_{t}-\alpha ^{*}\left( \frac{t}{T}\right)\right) \end{aligned}$$

(30)

and

$$\begin{aligned} \hat{\sigma }_{2,W}^{2}&= \frac{1}{T-h-W}\sum \limits _{t=W+1}^{T-h}\left( y_{T,t+h}-\beta ^{\prime }\left(\frac{t}{T}\right)z_{T,t}\right)^{2} \nonumber \\&\quad -\left( \frac{2}{T-h-W}\right) \sum \limits _{t=W+1}^{T-h}\left( y_{T,t+h}-\beta ^{\prime }\left(\frac{t}{T}\right)z_{T,t}\right) z_{T,t}^{\prime }\left( \widehat{\beta }_{t}-\beta \left(\frac{t}{T}\right) \right) \nonumber \\&\quad +\frac{1}{T-h-W} \sum \limits _{t=W+1}^{T-h} \left(\widehat{\beta } _{t}^{\prime }-\beta ^{\prime }\left(\frac{t}{T}\right)\right)^{\prime } z_{t}z_{t}^{\prime }\left( \widehat{\beta }_{t}-\beta \left(\frac{t}{T} \right) \right) , \end{aligned}$$

(31)

respectively. If we show that each of

$$\begin{aligned}&\frac{1}{T-h-W}\sum \limits _{t=W+1}^{T-h}\left( y_{T,t+h}-\alpha ^{*}\left(\frac{t}{T}\right)^{\prime }x_{T,t}\right) x_{t}^{\prime }\left( \widehat{\alpha }_{t}-{\alpha ^{*}\left(\frac{t}{T}\right)}\right)\nonumber \\&\quad - \frac{1}{T-h-W}\sum \limits _{t=W+1}^{T-h}u_{T,t+h}x_{T,t}^{\prime } B_{1}(t)H_{1}(t), \end{aligned}$$

(32)

$$\begin{aligned}&\frac{1}{T-h-W}\sum \limits _{t=W+1}^{T-h}\left( \widehat{\alpha } _{t}^{\prime }-\alpha ^{*}\left(\frac{t}{T}\right)\right)^{ \prime }x_{T,t}x_{T,t}^{\prime }\left( \widehat{\alpha }_{t}-\alpha ^{*}\left(\frac{t}{T}\right)\right) \nonumber \\&\quad - \frac{1}{T-h-W}\sum \limits _{t=W+1}^{T-h}H_{1}(t)^{\prime }B_{1}(t)z_{T,t}z_{T,t}^{\prime }B_{2}(t)H_{2}(t), \end{aligned}$$

(33)

$$\begin{aligned}&\frac{1}{T-h-W}\sum \limits _{t=W+1}^{T-h}\left(y_{T,t+h}-\beta \left( \frac{t}{T}\right)^{\prime }z_{T,t}\right)z_{T,t}^{\prime }\left( \widehat{ \beta }_{t}-\beta \left(\frac{t}{T}\right)\right) \nonumber \\&\quad - \frac{1}{T-h-W}\sum \limits _{t=W+1}^{T-h}v_{T,t+h}z_{T,t}^{\prime } B_{2}(t)H_{2}(t), \end{aligned}$$

(34)

$$\begin{aligned}&\frac{1}{T-h-W}\sum \limits _{t=W+1}^{T-h}\left( \widehat{\beta } _{t}^{\prime }-\beta \left(\frac{t}{T}\right)^{\prime }\right)^{\prime } z_{T,t}z_{T,t}^{\prime }\left( \widehat{\beta }_{t}-\beta \left(\frac{t}{T} \right)\right) \nonumber \\&\quad - \frac{1}{T-h-W}\sum \limits _{t=W+1}^{T-h}H_{2}(t)^{\prime }B_{2}(t)z_{T,t}z_{T,t}^{\prime }B_{2}(t)H_{2}(t), \end{aligned}$$

(35)

are $o_{p}(1/W)$ when the data are generated from model 1 (case 1) and are $o_{p}(1)$ when the data are generated from model 2 (case 2), the proof of Theorem 2 takes exactly the same steps as the proof of Theorem 1. Thus, it remains to show that (A.13)–(A.16) are $o_{p}(W^{-1})$ in case 1 and $o_{p}(1)$ in case 2. Note that the bias of the rolling regression estimator can be written as:

$$\begin{aligned} \hat{\beta }_{W,t}-\beta \left(\frac{t}{T}\right)&= B_{2}(t)\frac{1}{W_{h}} \sum _{s=t-W}^{t-h}z_{s} \left[v_{s+h}+z_{s}^{\prime }\left(\beta \left(\frac{s }{T}\right)-\beta \left(\frac{t}{T}\right)\right) \right] \nonumber \\&= B_{2}(t)H_{2}(t)+\frac{B_{2}(t)}{W_{h}}\sum _{s=t-W}^{t-h}z_{s}z_{s}^{ \prime }\left(\beta \left(\frac{s}{T}\right)-\beta \left(\frac{t}{T} \right)\right) \end{aligned}$$

(36)

Thus, the difference (A.15) is

$$\begin{aligned}&\frac{1}{T-h-W}\sum _{t=W+1}^{T-h}v_{T,t+h}z_{T,t}^{\prime }B_{2}(t) \frac{1}{W_{h}}\sum _{s=t-W}^{t-h}z_{s}z_{s}^{\prime }\left(\beta \left(\frac{s }{T}\right)-\beta \left(\frac{t}{T}\right)\right). \nonumber \\&= \frac{1}{T-h-W}\sum _{t=W+1}^{T-h}v_{T,t+h}z_{T,t}^{\prime }\bar{B} _{2}\left(\frac{t}{T}\right) \frac{1}{W_{h}}\sum _{s=t-W}^{t-h}z_{s}z_{s}^{ \prime }\left(\beta \left(\frac{s}{T}\right)-\beta \left(\frac{t}{T} \right)\right) \nonumber \\&\quad +\frac{1}{T-h-W}\sum _{t=W+1}^{T-h}v_{T,t+h}z_{T,t}^{\prime }\left(B_{2}(t)-\bar{B}_{2}\left(\frac{t}{T}\right)\right) \frac{1}{W_{h}} \nonumber \\&\quad \times \sum _{s=t-W}^{t-h}z_{s}z_{s}^{\prime } \left(\beta \left(\frac{s}{T} \right)-\beta \left(\frac{t}{T}\right)\right). \end{aligned}$$

(37)

By Assumption 4(c), the summands have zero mean. By Hölder’s inequality and Assumptions 4(b)(c)(e)(f), the second moments of the right-hand side terms are $O(W/T^{2})$. By Chebyshev’s inequality, (A.15) is $O_{p}(W^{1/2}/T)$ which is $o_{p}(1/W)$ by Assumption 3. It can be shown that (A.13) is also $o_{p}(1/W)$ in a similar fashion.

The difference (A.16) is the sum of the following three terms:

$$\begin{aligned} \frac{1}{T-h-W}\sum _{t=W+1}^{T-h} v_{T,t+h}z_{T,t}z_{T,t}^{\prime }B_{2}(t)\frac{1}{W_{h}}\sum _{s=t-W}^{t-h}z_{s}z_{s}^{\prime }\left(\beta \left(\frac{s}{T}\right)-\beta \left(\frac{t}{T}\right)\right), \end{aligned}$$

(38)

$$\begin{aligned} \frac{1}{T-h-W}\sum _{t=W+1}^{T-h} \left(\beta \left(\frac{s}{T} \right)-\beta \left(\frac{t}{T}\right)\right)^{\prime }\frac{1}{W_{h}} \sum _{s=t-W}^{t-h}z_{s}z_{s}^{\prime }B_{2}(t)z_{T,t}z_{T,t}^{\prime }v_{T,t+h}, \end{aligned}$$

(39)

$$\begin{aligned}&\frac{1}{T-h-W}\sum _{t=W+1}^{T-h} \left(\beta \left(\frac{s}{T} \right)-\beta \left(\frac{t}{T}\right)\right)^{\prime }\frac{1}{W_{h}} \sum _{s=t-W}^{t-h}z_{s}z_{s}^{\prime }B_{2}(t)z_{T,t} \nonumber \\&\quad \times z_{T,t}^{\prime }B_{2}(t)\frac{1}{W_{h}} \sum _{s=t-W}^{t-h}z_{s}z_{s}^{\prime }\left(\beta \left(\frac{s}{T} \right)-\beta \left(\frac{t}{T}\right)\right), \end{aligned}$$

(40)

Using Chebyshev’s inequality, Hölder’s inequality, Assumptions 3 and 4(b)(c)(e)(f), it can be shown that (A.19), (A.20), and (A.21) are $O_{p}(W^{1/2}T^{-2})$, $O_{p}(W^{1/2}T^{-2})$ and $O_{p}(W^{2}T^{-2})$ all of which are $o_{p}(W^{-1})$. It can be shown that (A.14) is also $o_{p}(1/W)$ when $\gamma (\cdot )=0$ in an analogous fashion.

The rest of the proof of Theorem 2 takes exactly the same steps as the proof of Theorem 1 except that $\alpha ^{*}$, $\beta$, $B_{i}$, $u_{t}$, $v_{t}$, $x_{t}$, $y_{t}$, $z_{t}$ and Lemma 1 is replaced by $\alpha \left(\frac{t}{T}\right)$, $\beta \left(\frac{t}{T}\right)$, $\bar{B }_{i}\left(\frac{t}{T}\right)$, $u_{Tt}$, $v_{Tt}$, $x_{Tt}$, $y_{Tt}$, $z_{Tt}$ and Lemma 2, respectively.