Bayesian Local Likelihood Estimation of Time-Varying DSGE Models: Allowing for Indeterminacy

Abstract

This paper modifies and employs a Bayesian Local Likelihood approach to estimate time-varying parameters of a New Keynesian model and assess such time variations using US data. Our modification contributes to the expanding literature by novelly integrating indeterminacy into the time-varying estimator. Further, we implement a one-step approach based on a unified solution set obtained by the augmented linearized rational expectation model simultaneously allowing for both determinacy and indeterminacy. The evidences suggest substantial time-variations in many parameters, particularly those associated with the Fed monetary policy rule and characterized by volatilities in the economy. This study also shows that allowing time-varying parameters improves density and point forecasts in comparison to a fixed-parameter DSGE model. We investigate implications of the time-variation for monetary policy effectiveness and find that the increase in the policy response to inflation from the pre-1979 to the post-1982 alone does not suffice for explaining the U.S. economy’s shift to determinacy, unless it is accompanied by either the estimated decline in trend inflation or the estimated change in policy responses to the output growth.

Appendices

Appendix A

The New Keynesian model in the main body can be represented as the standard Sims (2001) form as follows.

$$\begin{aligned} \Gamma _0(\theta )\textbf{X}_t=\Gamma _1(\theta )\textbf{X}_{t-1}+\Psi (\theta )\epsilon _t+\Pi (\theta )\eta _t \end{aligned}$$

The specific matrices have relevant forms below.

$$\begin{aligned}{} & {} \Gamma _0(\theta )\equiv \begin{bmatrix} 1 &{} 0 &{} \tau &{} -1 &{} -\tau &{} -1 &{} 0 \\ \kappa &{} -1 &{} 0 &{} 0 &{} \beta &{} 0 &{} -\kappa \\ \left( 1-\rho _R\right) \psi _2 &{} \left( 1-\rho _R\right) \psi _1 &{} -1 &{} 0 &{} 0 &{} 0 &{} -\left( 1-\rho _R\right) \psi _2 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 \\ 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \end{bmatrix} \\{} & {} \Gamma _1(\theta )\equiv \begin{bmatrix} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} -\rho _i &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \rho _g &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \rho _z \\ 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 \\ \end{bmatrix},\quad \Psi (\theta )\equiv \begin{bmatrix} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \\ -1 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 1 \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \\ \end{bmatrix},\quad \Pi (\theta )\equiv \begin{bmatrix} 0 &{} 0 \\ 0 &{} 0 \\ 0 &{} 0 \\ 0 &{} 0 \\ 0 &{} 0 \\ 1 &{} 0 \\ 0 &{} 1 \\ \end{bmatrix} \end{aligned}$$

Appendix B

Following Petrova (2019), we adopt the BLL estimation methodology to obtain estimates of time-varying parameters in DSGE models. We start with a brief outline description of this method. Conditionally on a batch of data $\textbf{D}^T=\left( \textbf{D}_1,\cdots ,\textbf{D}_T\right)$, Galvão et al. (2015) show that if time-varying parameters $\theta _t$ satisfy the condition

$$\begin{aligned} \underset{j:|j-t|\le h}{\sup }\begin{Vmatrix}\theta _t-\theta _j\end{Vmatrix}^2=O_p\left( \frac{h}{t}\right) ,\text { for }t\rightarrow \infty ,1\le h \le t, \end{aligned}$$

(20)

then the conditional log-density function ${l}\left( \textbf{D}_j|F_{j-1}\right)$ is only dependent on the parameter values in period j and the information set $F_{j-1}$ collected at the j-th time-point.

$$\begin{aligned} l\left( \textbf{D}_j|F_{j-1}\right) =l\left( \textbf{D}_j|\textbf{D}^{j-1},\theta _j\right) \end{aligned}$$

where $\textbf{D}^{j-1}\equiv (\textbf{D}_1,\textbf{D}_2,...,\textbf{D}_{j-1})$. Observed series $\left\{ \textbf{D}_t\right\} _{t=1}^T$ can be used to construct the sample pseudo-likelihood function:

$$\begin{aligned} L_t\left( \textbf{D}^T|\theta _t\right) =\prod _{j=1}^{T}L\left( \textbf{D}_t|\textbf{D}^{j-1},\theta _t\right) ^{w_{tj}},\text { for }t=1,...,T, \end{aligned}$$

(21)

where the weights are computed by the kernel function:

$$\begin{aligned} w_{tj}=K\left( \frac{t-j}{H}\right) , \text { for }t,j=1,...,T, \end{aligned}$$

where H is a bandwidth parameter to be selected appropriately. For estimating time-varying parameters, we choose a Gaussian kernel function in (22) to calculate the weight matrix, where the bandwidth parameter H is set equal to $T^{0.5}$. Then, the kernel function is

$$\begin{aligned} w_{tj}=\left( \frac{1}{\sqrt{2\pi }}\right) \exp \left\{ \left( -\frac{1}{2}\right) \left( \frac{t-j}{H}\right) ^2\right\} \end{aligned}$$

(22)

The kernel approach above is extended to Bayesian inference by Petrova (2019), who shows that the resulting quasi-posterior distributions are asymptotically Gaussian and valid for constructing confidence intervals as long as the weights $\tilde{w}_{tj}$ are modified as follows:

$$\begin{aligned} w_{tj}=\left( \sum _{j=1}^{T}\tilde{w}_{tj}^2\right) ^{-1}\left( \frac{\tilde{w}_{tj}}{\sum _{j=1}^{T}\tilde{w}_{tj}}\right) ,\text { for }j,t=1,...,T. \end{aligned}$$

One of the common methodologies for estimating $\theta _t$ in Equation (2) and (19) is to employ the implied one-step ahead prediction distribution $p_{\theta _t}\left( \textbf{D}_t|\textbf{D}_{t-1}\right)$ computed in (21) to form the likelihood of the observed series. Therefore, the information set $\textbf{D}^{t-1}$ can be simplified into $\textbf{D}_{t-1}$. Then the estimates of $\theta _t$ are yielded by maximizing the time-varying posterior probability function which, by Bayesian rule, can be represented as the product of the local pseudo-likelihood in period t and the prior probability of parameters.

$$\begin{aligned} \begin{aligned} \pi (\theta _t)&\equiv p_t\left( \theta _t|\textbf{D}^T\right) \\&=\frac{L_t\left( \textbf{D}^T|\theta _t\right) p\left( \theta _t\right) }{p\left( \textbf{D}^T\right) } \propto \prod _{j=1}^{T}L \left( \textbf{D}_j|\textbf{D}_{j-1},\theta _t\right) ^{w_{tj}}p\left( \theta _t\right) \end{aligned} \end{aligned}$$

For convenience, the formula above usually is taken in logarithms:

$$\begin{aligned} \log p_t\left( \theta _t|\textbf{D}^T\right) =-\sum _{j=1}^{T} w_{tj}\log L\left( \textbf{D}_t|\textbf{D}_{j-1},\theta _t\right) -\log p\left( \theta _t\right) . \end{aligned}$$

(23)

The standard Kalman filter methodology is adequate to compute $L_j\left( \textbf{D}_j|\textbf{D}_{j-1},\theta _t\right)$ for the purpose of evaluating the log-likelihood function $L_t\left( \textbf{D}^T|\theta _t\right)$. However, this posterior is intractable as the likelihood cannot be computed in closed form. Hence, we use the Random Walk Metropolis-Hastings (RWMH) algorithm, which is one of the most widely employed standard approaches among Markov Chain Monte Carlo (MCMC) techniques to evaluate (23) numerically. The RWMH proceeds by selecting an arbitrary proposal Markov Chain $q\left( \theta _t^i|\theta _t^{i-1}\right)$ and moving it forward by a way of randomly accepted draws from the chain at the last iterate. The acceptance probability is dependent on the posterior distribution. In the majority of Bayesian contexts, a Gaussian proposal distribution is used,

$$\begin{aligned} q\left( \theta _t^{(i)}|\theta _t^{(i-1)}\right) =\mathcal {N}\left[ \theta _t';\mu \left( \theta _t^{(i-1)}\right) ,\mathbf {\Sigma }\left( \theta _t^{(i-1)}\right) \right] , \end{aligned}$$

where $\theta _t'$ are the candidate parameter estimates (represented as a swarm of particles) with $\mu$ and $\mathbf {\Sigma }$ denoting the mean and covariance matrix of the joint Gaussian distribution, respectively. The Gaussian mean, obtained by a maximization routine, is the mode of local posterior distribution, which is different during every period.

Once the inverse Hessian matrix of log likelihood is evaluated at posterior modes $\mu \left( \theta _t\right) =\hat{\theta }_t$, the variance $\mathbf {\Sigma }\left( \hat{\theta }_t \right)$ of the proposal densities of parameters can be obtained. Afterwards, parameter initial values $\theta _t^0$ can be drawn from the joint Gaussian distribution,

$$\begin{aligned} \mathcal {N}\left[ \theta _t^0;\hat{\theta _t},c_0^2\mathbf {\Sigma }\left( \hat{\theta _t}\right) \right] . \end{aligned}$$

After the initial step, the MH recursive steps proceed. For $i=1,...,n_{sim}$, we draw particles $\theta _t^{(i)}$ from the proposal Gaussian distribution,

$$\begin{aligned} \mathcal {N}\left[ \theta _t';\theta _t^{(i-1)},c^2\mathbf {\Sigma }\left( \hat{\theta _t}\right) \right] \end{aligned}$$

(24)

With respect to each draw, the covariance matrix of the Gaussian distribution is invariant. After generating a candidate parameter set via (24), it is accepted in the acceptance probability as follows,

$$\begin{aligned} \alpha \left( \theta _t',\theta _t^{(i-1)}\right) =\textbf{1}\wedge \frac{\pi (\theta _t')q\left( \theta _t^{(i-1)}|\theta _t'\right) }{\pi \left( \theta _t^{(i-1)}\right) q\left( \theta _t'|\theta _t^{(i-1)}\right) } \end{aligned}$$

where $a\wedge b=\min (a,b)$. We set $\theta _t^{(i)}\leftarrow \theta _t'$ if the candidate parameter particle set is accepted and $\theta _t^{(i)}\leftarrow \theta _t^{(i-1)}$ if it is rejected. $c_0^2$ and $c^2$ are scaling parameters that tune the step length of the RWMH algorithm. Tuning them, we aim to keep acceptance probability within the range 25–35%.

Appendix C

1.1 The Point Forecasts are Computed as Posterior Means

$$\begin{aligned} \hat{\textbf{D}}_{T+h|T}=\int _{\textbf{D}_{T+h}}\textbf{D}_{T+h}\times p\left( \textbf{D}_{T+h}|\textbf{D}_{1:T}\right) \textbf{d}\textbf{D}_{T+h}\approx \frac{1}{n_{sim}}\sum _{j=1}^{n_{sim}}\textbf{D}_{T+h}^{(j)} \end{aligned}$$

(25)

where the draws $\textbf{D}_{T+h}^{(j)}$ are obtained using Algorithm 1.

1.2 Mean Forecast Bias

Based on forecast estimates of Eq. (25), the expectation biases can be computed as follows.

$$\begin{aligned} \textbf{Bias}\left( n|h\right) =\frac{1}{p-h}\sum _{T=E}^{E+p-h}\left( \textbf{D}_{n,T+h}-\hat{\textbf{D}}_{n,T+h|T}\right) \end{aligned}$$

(26)

where $\textbf{D}_{n,T+h}$ is actually observed values at time $T+h$; $\hat{\textbf{D}}_{n,T+h|T}$ denote h-steps forward expectations; p is the number of forecasting rounds, and n represents variables y, $\pi$ or i, respectively.

The root mean squared error (RMSE) is computed as follows,

$$\begin{aligned} \textbf{RMSE}(n|h)=\sqrt{\frac{1}{p-h}\sum _{T=E}^{E+p-h}\left( \textbf{D}_{n,T+h}-\hat{\textbf{D}}_{n,T+h|T}\right) ^2} \end{aligned}$$

(27)

where E is the first forecast origin and p stands for the number of h-step forecasting rounds.

1.3 Log Predictive Scores

The approach of log predictive scores suggested by Good (1992) is popularly used in the literature. Scoring rules can be used to compare the ability of forecasts between a collection of models by assigning the predictive distribution a specified value that materializes at observations. Here, we report the differences between the log predictive scores computed from the time-varying DSGE model and those computed from the fixed-parameter DSGE model, and also report the differences between the log predictive scores from the TV-DSGE model and those from the TV-BVAR model. If relative log predictive scores are larger than zero, it means that the time-varying parameter DSGE model is comparatively better for prediction. The log predictive scores can be computed as the summation of log marginal predictive likelihood:

$$\begin{aligned} \textbf{S}(n|h)=\frac{1}{p-h}\sum _{T=E}^{E+p-h}\log p\left( \textbf{D}_{n,T+h}|\textbf{D}_{n,1:T}\right) \end{aligned}$$

(28)

1.4 Probability Integral Transform (PIT)

The log predictive scores rank predictive accuracies according to estimated likelihoods of the realized observables. As is well-known, a well-calibrated density forecast should satisfy two criteria: unbiasedness and efficiency. Equipped with this idea, we turn to another tool for comparing the density forecasts, that is, the probability integral transform, which can be used to assess the extent to which these two criteria are satisfied. According to Diebold et al. (2017), the probability integral transform is computed as the cumulative probability of the random variable $\textbf{D}_{n,T+h}$ evaluated at the ex-post realizations of the variables collected in $\textbf{D}_{n,T+h}$,

$$\begin{aligned} \textbf{PIT}_{n,h,T}=\int _{-\infty }^{\textbf{D}_{n,T+h}}p\left( \textbf{D}_{n,T+h}|\textbf{D}_{n,1:T}\right) \textbf{d}\textbf{D}_{n,T+h} \end{aligned}$$

The formula above is based on the predictive density function evaluated at the observed values until the $T^{th}$ time point. In practice, the probability integral transform can be obtained by the Monte Carlo average approach as follows:

$$\begin{aligned} \textbf{PIT}_{n,h,T}\approx \frac{1}{n_{sim}}\sum _{j=1}^{n_{sim}}\textbf{I}\left( \textbf{D}_{n,T+h}^{(j)}\le \textbf{D}_{n,T+h}\right) \end{aligned}$$

(29)

Appendix D

We consider the following VAR(2) model with time-varying parameters

$$\begin{aligned} \begin{aligned} \textbf{Y}_t&=c_t + \sum _{j=1}^2B_{j,t}\textbf{Y}_{t-j}+\nu _t,\quad \textbf{VAR}(\nu _t)=R_t\\ \beta _t&=\{c_t,B_{1,t},B_{2,t}\} \\ \beta _t&=\beta _{t-1}+e_t,\quad \textbf{VAR}(e_t)=Q \end{aligned} \end{aligned}$$

The covariance matrix of the error term $\nu _t$, i.e. $R_t$ has time-varying elements. The observable vector $\textbf{Y}_t=(y_t,\pi _t,i_t)$, every element being output growth, inflation and interest rate, respectively. For simplicity we consider the following structure for $R_t$,

$$\begin{aligned} R_t=A_t^{-1}H_tA_t^{-1'} \end{aligned}$$

where $A_t$ is a lower triangular matrix and $H_t$ is a diagonal matrix. For a three variable VAR above,

$$\begin{aligned} A_t= \begin{bmatrix} 1 &{} 0 &{} 0 \\ \alpha _{12,t} &{} 1 &{} 0 \\ \alpha _{13,t}&{} \alpha _{23,t} &{} 1 \end{bmatrix},\quad H_t= \begin{bmatrix} h_{1,t} &{} 0 &{} 0 \\ 0 &{} h_{2,t} &{} 0 \\ 0&{} 0 &{} h_{3,t} \end{bmatrix} \end{aligned}$$

where

$$\begin{aligned} \alpha _{ij,t}=\alpha _{ij,t-1}+V_t,\quad \textbf{VAR}(V_t)=D \end{aligned}$$

and

$$\begin{aligned} \ln h_{i,t}=\ln h_{i,t-1}+z_{i,t},\quad \textbf{VAR}(z_{i,t})=g_i \end{aligned}$$

for $i=1,2,3$.

The Gibbs and MH algorithm for estimating this three variable time-varying VAR model. The sample is the same as what has been used in estimating the New Keynesian model. We choose the first 40 quarterly data as warm-up sample to estimate a three- variable VAR(2) model. The estimates function as initial values when estimating the TV-BVAR model. We implement MH algorithm with a number of draws of 370,000, from which we drop the first about 10% draws.

Appendix E

The paper employs three U.S. quarterly time series: GDP growth rate $100\Delta \log GDP_t$, quarterly growth rate of the urban CPI index $100\Delta \log CPI_t$ and the Federal Funds rate $100R_t/4$. To compare the monetary policy effects to the economy, it divides the sample over two subsample periods. The first subsample, 1948:Q1–1979:Q4, corresponds to the Great Inflation period. The second one, 1980:Q1–2019:Q4, corresponds to the Great Moderation period characterized by dramatically milder macroeconomic volatilities. The measurement equations relating the relevant elements of to the three observables are given by (9) in the main body. The raw data should be processed such that we estimate the model to match them.

Firstly, Growth rate of real output per capita.

$$\begin{aligned} YGR_t=100\left[ \ln \left( GDP_t\right) -\ln \left( GDP_{t-1}\right) \right] \end{aligned}$$

Secondly, Annualized inflation rate.

$$\begin{aligned} INFL_t=400\ln \left( \frac{CPI_t}{CPI_{t-1}}\right) \end{aligned}$$

Thirdly the Fed funds rate

$$\begin{aligned} INT_t=FFR_t/4 \end{aligned}$$