Zhongjun Qu

Long-Memory and Level Shifts in the Volatility of Stock Market Return Indices

There has been interest in the possibility of confusing long memory and structural changes in level, and studies showed that when a short-memory process is contaminated by level shifts the estimate of the fractional differencing parameter is biased upward and the autocovariances decay slowly. We analyze the properties of the autocorrelation function, the periodogram, and the log periodogram estimate of the memory parameter when the level shift component is a mixture model. Our results explain many findings reported and uncover new features. We confront our theoretical predictions using log-squared returns as a proxy for the volatility of some assets returns, including daily S&P 500 returns over the period 1928–2002. The estimates follow patterns that would obtain if the process was short memory with level shifts instead of fractionally integrated. A simple test is also proposed, which reinforces this conclusion.

A Test Against Spurious Long Memory

This paper proposes a test statistic for the null hypothesis that a given time series is a stationary long-memory process against the alternative hypothesis that it is affected by regime change or a smoothly varying trend. The proposed test is in the frequency domain and is based on the derivatives of the profiled local Whittle likelihood function in a degenerating neighborhood of the origin. The assumptions used are mild, allowing for non-Gaussianity or conditional heteroscedasticity. The resulting null limiting distribution is free of nuisance parameters and can be easily simulated. Furthermore, the test is straightforward to implement; in particular, it does not require specifying the form of the trend or the number of different regimes under the alternative hypothesis. Monte Carlo simulation shows that the test has decent size and power properties. The article also considers three empirical applications to illustrate the usefulness of the test. This article has supplementary material online.

Testing for Structural Change in Regression Quantiles

Most studies in the structural change literature focus solely on the conditional mean, while under various circumstances, structural change in the conditional distribution or in conditional quantiles is of key importance. This paper proposes several tests for structural change in regression quantiles. Two types of statistics are considered, namely, a fluctuation type statistic based on the subgradient and a Wald type statistic, based on comparing parameter estimates obtained from different subsamples. The former requires estimating the model under the null hypothesis, and the latter involves estimation under the alternative hypothesis. The tests proposed can be used to test for structural change occurring in a pre-specified quantile, or across quantiles, which can be viewed as testing for change in the conditional distribution with a linear specification of the conditional quantile function. Both single and multiple structural changes are considered. We derive the limiting distributions under the null hypothesis, and show they are nuisance parameter free and can be easily simulated. A simulation study is conducted to assess the size and power in finite samples.

Identification and frequency domain quasi-maximum likelihood estimation of linearized dynamic stochastic general equilibrium models

This paper considers issues related to identification, inference, and computation in linearized dynamic stochastic general equilibrium (DSGE) models. We first provide a necessary and sufficient condition for the local identification of the structural parameters based on the (first and) second order properties of the process. The condition allows for arbitrary relations between the number of observed endogenous variables and structural shocks, and is simple to verify. The extensions, including identification through a subset of frequencies, partial identification, conditional identification, and identification under general nonlinear constraints, are also studied. When lack of identification is detected, the method can be further used to trace out nonidentification curves. For estimation, restricting our attention to nonsingular systems, we consider a frequency domain quasi-maximum likelihood estimator and present its asymptotic properties. The limiting distribution of the estimator can be different from results in the related literature due to the structure of the DSGE model. Finally, we discuss a quasi-Bayesian procedure for estimation and inference. The procedure can be used to incorporate relevant prior distributions and is computationally attractive.

A Stochastic Volatility Model with Random Level Shifts: Theory and Applications to S&P 500 and NASDAQ Return Indices

Empirical ?ndings related to the time series properties of stock returns volatility indicate autocorrelations that decay slowly at long lags. In light of this, several long-memory models have been proposed. However, the possibility of level shifts has been advanced as a possible explanation for the appearance of long-memory and there is growing evidence suggesting that it may be an important feature of stock returns volatility. Nevertheless, it remains a conjecture that a model incorporating random level shifts in variance can explain the data well and produce reasonable forecasts. We show that a very simple stochastic volatility model incorporating both a random level shift and a short-memory component indeed provides a better in-sample fit of the data and produces forecasts that are no worse, and sometimes better, than standard stationary short and long-memory models. We use a Bayesian method for inference and develop algorithms to obtain the posterior distributions of the parameters and the smoothed estimates of the two latent components. We apply the model to daily S&P 500 and NASDAQ returns over the period 1980.1-2005.12. Although the occurrence of a level shift is rare, about once every two years, the level shift component clearly contributes most to the total variation in the volatility process. The half-life of a typical shock from the short-memory component is very short, on average between 8 and 14 days. We also show that, unlike common stationary short or long-memory models, our model is able to replicate keys features of the data. For the NASDAQ series, it forecasts better than a standard stochastic volatility model, and for the S&P 500 index, it performs equally well.

A Modified Information Criterion for Cointegration Tests based on a VAR Approximation

We consider Johansen’s (1988, 1991) cointegration tests when a Vector AutoRegressive (VAR) process of order k is used to approximate a more general linear process with an infinite VAR representation. In this case, and in particular when a moving average component is present, traditional methods to select the lag order, such as Akaike’s (AIC) or the Bayesian information criteria, lead to too parsimonious a model, with the implication that the cointegration tests suffer from substantial size distortions in finite samples. We extend the analysis of Ng and Perron (2001) to derive a Modified Akaike’s Information Criterion (MAIC) in this multivariate setting. The idea is to use the information specified by the null hypothesis as it relates to restrictions on the parameters of the model to keep an extra term in the penalty function of the AIC. This MAIC takes a very simple form for which this extra term is simply the likelihood ratio test for testing the null hypothesis of r against more than r cointegrating vectors. We provide theoretical analyses of its validity and of the fact that cointegration tests constructed from a VAR whose lag order is selected using the MAIC have the same limit distribution as when the order is finite and known. We also provide theoretical and simulation analyses to show how the MAIC leads to VAR approximations that yield tests with drastically improved size properties with little loss of power.

Searching for cointegration in a dynamic system

, which can be zero, stable cointegrating vectors against the alternative hypothesis of more than r 0 cointegrating vectors existing in some subsample. The tests proposed follow Breitung (2002). They are non-parametric in nature and are invariant to linear transformations of the series. A distinctive feature is that they allow us to detect the hidden cointegration when the system is affected by an unknown number of regime changes of unknown timing. We analyse the limiting distributions and provide tables of critical values. Various extensions are then discussed which incorporate a priori information to improve the power. A simple correction is also proposed to yield improved finite sample performance. Finally simulations are conducted to evaluate the size and power in finite samples. Copyright Royal Economic Society 2007

Inference in dynamic stochastic general equilibrium models with possible weak identification

This paper considers inference in log‐linearized dynamic stochastic general equilibrium (DSGE) models with weakly (including un‐) identified parameters. The framework allows for analysis using only part of the spectrum, say at the business cycle frequencies. First, we characterize weak identification from a frequency domain perspective and propose a score test for the structural parameter vector based on the frequency domain approximation to the Gaussian likelihood. The construction heavily exploits the structures of the DSGE solution, the score function, and the information matrix. In particular, we show that the test statistic can be represented as the explained sum of squares from a complex‐valued Gauss–Newton regression, where weak identification surfaces as (imperfect) multicollinearity. Second, we prove that asymptotically valid confidence sets can be obtained by inverting this test statistic and using chi‐squared critical values. Third, we provide procedures to construct uniform confidence bands for the impulse response function, the time path of the variance decomposition, the individual spectrum, and the absolute coherency. Finally, a simulation experiment suggests that the test has adequate size even with relatively small sample sizes. It also suggests it is possible to have informative confidence sets in DSGE models with unidentified parameters, particularly regarding the impulse response functions. Although the paper focuses on DSGE models, the methods are applicable to other dynamic models with well defined spectra, such as stationary (factor‐augmented) vector autoregressions.

Frequency Domain Analysis of Medium Scale DSGE Models with Application to Smets and Wouters (2007)

The chapter considers parameter identification, estimation, and model diagnostics in medium scale DSGE models from a frequency domain perspective using the framework developed in Qu and Tkachenko (2012). The analysis uses Smets and Wouters (2007) as an illustrative example, motivated by the fact that it has become a workhorse model in the DSGE literature. For identification, in addition to checking parameter identifiability, we derive the non-identification curve to depict parameter values that yield observational equivalence, revealing which and how many parameters need to be fixed to achieve local identification. For estimation and inference, we contrast estimates obtained using the full spectrum with those using only the business cycle frequencies to find notably different parameter values and impulse response functions. A further comparison between the nonparametrically estimated and model implied spectra suggests that the business cycle based method delivers better estimates of the features that the model is intended to capture. Overall, the results suggest that the frequency domain based approach, in part due to its ability to handle subsets of frequencies, constitutes a flexible framework for studying medium scale DSGE models.

M Tests with a New Normalization Matrix

This paper proposes a new family of M tests building on the work of Kuan and Lee (2006) and Kiefer et al. (2000). The idea is to replace the asymptotic covariance matrix in conventional M tests with an alternative normalization matrix, constructed using moment functions estimated from (K + 1) recursive subsamples. The new tests are simple to implement. They automatically account for the effect of parameter estimation and allow for conditional heteroskedasticity and serial correlation of general forms. They converge to central F distributions under the fixed-K asymptotics and to chi-square distributions if K is allowed to approach infinity. We illustrate their applications using three simulation examples: (1) specification testing for conditional heteroskedastic models, (2) non-nested testing with serially correlated errors, and (3) testing for serial correlation with unknown heteroskedasticity. The results show that the new tests exhibit good size properties with power often comparable to the conventional M tests while being substantially higher than that of Kuan and Lee (2006).