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Abstract:
We compare 330 ARCH-type models in terms of their ability to describe the conditional variance. The models are compared out-of-sample using DM-$ exchange rate data and IBM return data, where the latter is based on a new data set of realized variance. We find no evidence that a GARCH(1,1) is outperformed by more sophisticated models in our analysis of exchange rates, whereas the GARCH(1,1) is clearly inferior to models that can accommodate a leverage effect in our analysis of IBM returns. The models are compared with the test for superior predictive ability (SPA) and the reality check for data snooping (RC). Our empirical results show that the RC lacks power to an extent that makes it unable to distinguish 'good' and 'bad' models in our analysis.

Volatility Models, Forecast Comparison, Realized Variance, Superior Predictive Ability

Abstract:
We study market microstructure noise in high-frequency data and analyze its implications for the realized variance (RV) under a general specification for the noise. We show that kernel-based estimators can unearth important characteristics of marketmicrostructure noise and that a simple kernel-based estimator dominates the RV for the estimation of integrated variance (IV). An empirical analysis of the Dow Jones Industrial Average stocks reveals that market microstructure noise is time-dependent and correlated with increments in the efficient price. This has important implications for volatility estimation based on high-frequency data. Finally, we apply cointegration techniques to decompose transaction prices and bid-ask quotes into an estimate of the efficient price and noise. This framework enables us to study the dynamic effects on transaction prices and quotes caused by changes in the efficient price.

Realized variance, realized volatility, integrated variance, market microstructure noise, bias correction, high-frequency data, sampling schemes

Abstract:
We propose a new test for superior predictive ability. The new test compares favorable to the reality check for data snooping (RC), because the former is more powerful and less sensitive to poor and irrelevant alternatives. The improvements are achieved by two modifications of the RC. We employ a studentized test statistic that reduces the influence of erratic forecasts and invoke a sample dependent null distribution. The advantages of the new test are confirmed by Monte Carlo experiments and in an empirical exercise, where we compare a large number of regression-based forecasts of annual US inflation to a simple random walk forecast. The random walk forecast is found to be inferior to regression-based forecasts and, interestingly, the best sample performance is achieved by models that have a Phillips curve structure.

Testing for Superior Predictive Ability, Forecasting, Forecast Evaluation, Multiple Comparisons, Inequality Testing.

Abstract:
This paper studies tests of calendar effects in equity returns. It is necessary to control for all possible calendar effects to avoid spurious results. The authors contribute to the calendar effects literature and its significance with a test for calendar-specific anomalies that conditions on the nuisance of possible calendar effects. Thus, their approach to test for calendar effects produces robust data-mining results. Unfortunately, attempts to control for a large number of possible calendar effects have the downside of diminishing the power of the test, making it more difficult to detect actual anomalies. The authors show that our test achieves good power properties because it exploits the correlation structure of (excess) returns specific to the calendar effect being studied. We implement the test with bootstrap methods and apply it to stock indices from Denmark, France, Germany, Hong Kong, Italy, Japan, Norway, Sweden, the United Kingdom, and the United States. Bootstrap p- values reveal that calendar effects are significant for returns in most of these equity markets, but end-of-the-year effects are predominant. It also appears that, beginning in the late 1980s, calendar effects have diminished except in small-cap stock indices.

Calendar effects, data mining, significance test

Abstract:
This paper introduces the model confidence set (MCS) and applies it to the selection of models. An MCS is a set of models that is constructed so that it will contain the best model with a given level of confidence. The MCS is in this sense analogous to a confidence interval for a parameter. The MCS acknowledges the limitations of the data; uninformative data yield an MCS with many models whereas informative data yield an MCS with only a few models. The MCS procedure does not assume that a particular model is the true model; in fact, the MCS procedure can be used to compare more general objects, beyond the comparison of models. We apply the MCS procedure to two empirical problems. First, we revisit the inflation forecasting problem posed by Stock and Watson (1999) and compute the MCS for their set of inflation forecasts. Second, we compare a number of Taylor rule regressions and determine the MCS of the best in terms of in-sample likelihood criteria.

Model confidence set, forecasting, model selection, multiple comparisons

Abstract:
This paper shows how to use realised kernels to carry out efficient feasible inference on the ex-post variation of underlying equity prices in the presence of simple models of market frictions. The issue is subtle with only estimators which have symmetric weights delivering consistent estimators with mixed Gaussian limit theorems. The weights can be chosen to achieve the best possible rate of convergence and to have an asymptotic variance which is close to that of the maximum likelihood estimator in the parametric version of this problem. Realised kernels can also be selected to (i) be analysed using endogenously spaced data such as that in databases on transactions, (ii) allow for market frictions which are endogenous, (iii) allow for temporally dependent noise. The finite sample performance of our estimators is studied using simulation, while empirical work illustrates their use in practice.

Bipower variation, Long run variance estimator, Market frictions, Quadratic variation, Realized variance, Subsampling

Abstract:
The realized variance (RV) is known to be biased because intraday returns are contaminated with market microstructure noise, in particular if intraday returns are sampled at high frequencies. In this paper, we characterize the bias under a general specification for the market microstructure noise, where the noise may be autocorrelated and need not be independent of the latent price process. Within this framework, we propose a simple Newey-West type correction of the RV that yields an unbiased measure of volatility, and we characterize the optimal unbiased RV in terms of the mean squared error criterion. Our empirical analysis of the 30 stocks of the Dow Jones Industrial Average index shows the necessity of our general assumptions about the noise process. Further, the empirical results show that the modified RV is unbiased even if intraday returns are sampled every second.

Realized variance, realized volatility, high-frequency data, integrated variance

Abstract:
We consider the problem of deriving an empirical measure of daily integrated variance (IV) in the situation where high-frequency price data are unavailable for part of the day. We study three estimators in this context and characterize the assumptions that justify their use. We show that the optimal combination of the realized variance and squared overnight return can be determined, despite the latent nature of IV, and we discuss this result in relation to the problem of combining forecasts. Finally, we apply our theoretical results and construct four years of daily volatility estimates for the 30 stocks of the Dow Jones Industrial Average.

high-frequency data, market microstructure noise, realized variance

Abstract:
We consider the problem of deriving an empirical measure of daily integrated variance (IV) in the situation where high-frequency price data are unavailable for part of the day. We study three estimators in this context and characterize the assumptions that justify their use. We show that the optimal combination of the realized variance and squared overnight return can be determined, despite the latent nature of IV, and we discuss this result in relation to the problem of combining forecasts. Finally, we apply our theoretical results and construct four years of daily volatility estimates for the 30 stocks of the Dow Jones Industrial Average.

Realized Variance, High-Frequency Data, Market Microstructure Noise

Abstract:
This paper applies the model confidence sets (MCS) procedure to a set of volatility models. A MSC is analogous to a confidence interval of parameter in the sense that the former contains the best forecasting model with a certain probability. The key to the MCS is that it acknowledges the limitations of the information in the data. The empirical exercise is based on fifty-five volatility models, and the MCS includes about a third of these when evaluated by mean square error, whereas the MCS contains only a VGARCH model when mean absolute deviation criterion is used. We conduct a simulation study that shows the MCS captures the superior models across a range of significance levels. When we benchmark the MCS relative to a Bonferroni bound, this bound delivers inferior performance.

Forecasting, Model Selection, Multiple Comparisons, Data Mining

Abstract:
We show that the empirical ranking of volatility models can be inconsistent for the true ranking if the evaluation is based on a proxy for the population measure of volatility. For example, the substitution of a squared return for the conditional variance in the evaluation of ARCH-type models can result in an inferior model being chosen as "best" with a probability that converges to one as the sample size increases. We document the practical relevance of this problem in an empirical application and by simulation experiments. Our results provide an additional argument for using the realized variance in out-of-sample evaluations rather than the squared return. We derive the theoretical results in a general framework that is not specific to the comparison of volatility models. Similar problems can arise in comparisons of forecasting models whenever the predicted variable is a latent variable.

Consistent Ranking, Model Comparison, Volatility Models.

Abstract:
We introduce an estimation method that applies to a class of multivariate regression problems. The method can estimate parameters that are subject to multiple reduced-rank conditions and other parameter restrictions and the method allows for a general specifications of the covariance matrix. We refer to a regression problem of this type as a generalized reduced rank regression (GRRR). Several econometric estimation problems can be formulated as a GRRR. We illustrated this with examples, where the leading case is estimation of panel cointegration models.

Reduced Rank Regression, Generalized Reduced Rank Regression, Panel-Cointegration, Cointegration

Abstract:
The Granger representation theorem states that a cointegrated vector autoregressive process can be decomposed into the four components: a random walk, a stationary process, a deterministic part, and a term that depends on the initial values. In this paper, we present a new proof of the theorem. This proof enables us to derive closed-form expressions of all terms of the representation and allows a unified treatment of models with dierent deterministic specifications. The applicability of our results are illustrated by examples. For example, the closed-form expressions are useful for impulse response analyses and facilitate the analysis of cointegration models with structural changes.

Cointegration, Granger Representation, I(1), Impulse Response Analysis

Abstract:
This rejoinder clarifies issues related to the features of market microstructure noise. Specifically, we show that pre-processing of high-frequency data is very useful for the estimation of quadratic variation. We also document a strong relationship between quadratic variation and the number of transactions.

Realized Variance, Market Microstructure Noise, Pre-processing of High-Frequency data, outliers, sampling schemes

Abstract:
Realised kernels use high frequency data to estimate daily volatility of individual stock prices. They can be applied to either trade or quote data. Here we provide the details of how we suggest implementing them in practice. We compare the estimates based on trade and quote data for the same stock and find a remarkable level of agreement.

We identify some features of the high frequency data which are challenging for realised kernels. They are when there are local trends in the data, over periods of around 10 minutes, where the prices and quotes are driven up or down. These can be associated with high volumes. One explanation for this is that they are due to non-trivial liquidity effects.

HAC estimator, Long run variance estimator, Market frictions, Quadratic variation

Abstract:
Test statistics that are suitable for testing composite hypotheses are typically non-pivotal, and conservative bounds are commonly used to test composite hypotheses. In this paper, we propose a testing procedure for composite hypotheses that incorporates additional sample information. This avoids, as n->oo, the use of conservative bounds and leads to tests with better power than standard tests. The testing procedure satisfies a novel similarity condition that is relevant for asymptotic tests of composite hypotheses, and we show that this is a necessary condition for a test to be unbiased. The procedure is particularly useful for simultaneous testing of multiple inequalities, in particular when the number of inequalities is large.This is the situation for the multiple comparisons of forecasting models, and we show that the new testing procedure dominates the 'reality check' of White (2000) and avoids certain pitfalls.

Composite hypothesis, similar test, unbiased test, multiple comparisons

Abstract:
We propose a multivariate realised kernel to estimate the ex-post covariation of log-prices. We show this new consistent estimator is guaranteed to be positive semi-definite and is robust to measurement noise of certain types and can also handle non-synchronous trading. It is the first estimator which has these three properties which are all essential for empirical work in this area. We derive the large sample asymptotics of this estimator and assess its accuracy using a Monte Carlo study. We implement the estimator on some US equity data, comparing our results to previous work which has used returns measured over 5 or 10 minutes intervals. We show the new estimator is substantially more precise.

HAC estimator, Long run variance estimator, Market frictions, Quadratic variation, Realised Variance

Abstract:
We derive an identity for the determinant of a product involving non-squared matrices. The identity can be used to derive the maximum likelihood estimator in reduced-rank regressions with Gaussian innovations. Furthermore, the identity sheds light on the structure of the estimation problem that arises when the reduced-rank parameters are subject to additional constraints.

Determinant Identity, Reduced Rank Regression, Least Squares

Abstract:
We derive an identity for the determinant of a product involving non-squared matrices. The identity can be used to derive the maximum likelihood estimator in reduced-rank regressions with Gaussian innovations. Furthermore, the identity sheds light on the structure of the estimation problem that arises when the reduced-rank parameters are subject to additional constraints.

Determinant Identity, Reduced Rank Regression, Least Squares

Abstract:
In a recent paper we have introduced the class of realised kernel estimators of the increments of quadratic variation in the presence of noise. We showed this estimator is consistent and derived its limit distribution under various assumptions on the kernel weights. In this paper we extend our analysis, looking at the class of subsampled realised kernels and we derive the limit theory for this class of estimators. We find that subsampling is highly advantages for estimators based on discontinuous kernels, such as the truncated kernel. For kinked kernels, such as the Bartlett kernel, we show that subsampling is impotent, in the sense that subsampling has no effect on the asymptotic distribution. Perhaps surprisingly, for the efficient smooth kernels, such as the Parzen kernel, we show that subsampling is harmful as it increases the asymptotic variance. We also study the performance of subsampled realised kernels in simulations and in empirical work.

Bipower variation, Long run variance estimator, Market frictions, Quadratic variation, Realised kernel, Realised variance, Subsampling

Abstract:
We examine moving average (MA) filters for estimating the integrated variance (IV) of a financial asset price in a framework where high-frequency price data are contaminated with market microstructure noise. We show that the sum of squared MA residuals must be scaled to enable a suitable estimator of IV. The scaled estimator is shown to be consistent, first-order efficient, and asymptotically Gaussian distributed about the integrated variance under restrictive assumptions. Under more plausible assumptions, such as time-varying volatility, the MA model is misspecified. This motivates an extensive simulation study of the merits of the MA-based estimator under misspecification. Specifically, we consider nonconstant volatility combined with rounding errors and various forms of dependence between the noise and efficient returns. We benchmark the scaled MA-based estimator to subsample and realized kernel estimators and find that the MA-based estimator performs well despite the misspecification.

Bias correction, High-frequency data, Integrated variance, Moving average, Realized variance, Realized volatility

Abstract:
We examine moving average (MA) filters for estimating the integrated variance of a financial asset price in a framework where high frequency price data are contaminated with marketmicrostructure noise. We show that the sum of squared MA residuals needs to be scaled for it to be a suitable estimator of integrated variance. The scaled estimator is shown to be consistent, first-order efficient, and asymptotically Gaussian distributed about the integrated variance under restrictive assumptions. Under more plausible assumptions, such as timevarying volatility, the MA model is misspecified. This motivates an extensive simulation study of the merits of the MA-based estimator under mispecification. Specifically we consider: non-constant volatility combined with rounding errors and various forms of dependence between the noise and efficient returns. We benchmark the scaled MA-based estimator to subsample and realized kernel estimators and find that the MA-based estimator performs well despite the misspecification.

Integrated Variance, Realized Variance, Realized Volatility, Moving Average, Bias Correction

Abstract:
In this paper, we provide an intraday analysis of the impact of monetary policy on the equity markets. Specifically, we study changes in prices and changes in volatility for the S&P 500 associated with Federal Open Market Committee announcements as well as real-time changes in market expectations about future policy. The analysis shows an economically and statistically significant inverse relationship between equity market returns and changes in the Fed funds rate target. The magnitude of the response is dependent on whether the change was expected or unexpected. An expected change in the Fed funds rate target of 25 basis points results in approximately a 30 basis point decline in the broad equity market, while an unexpected change of 25 basis points in the Fed funds rate target results in approximately 125 basis point decline in the broad equity market. The speed of these market reactions is rapid with the equity market reaching a new equilibrium within fifteen minutes. In contrast to these results, the analysis also shows a positive relationship exists between equity market returns and changes in expectations about future monetary policy. Taken together, these results regarding price changes (returns) suggest that the price discovery process in the equity markets is dominated by the realization of expectations and not market expectations per se. Meanwhile, the volatility analysis suggests a volatility spike follows both FOMC announcements and real-time changes in expectations, but the duration of these spikes is relatively short-lived and dampens out within one hour.

Monetary Policy, Exchange Traded Funds, Realized Variance, High-Frequency Data

Abstract:
We introduce a novel estimator of the quadratic variation that is based on the theory of Markov chains. The estimator is motivated by some general results concerning filtering contaminated semimartingales. Specifically, we show that filtering can in principle remove the effects of market microstructure noise in a general framework where little is assumed about the noise. For the practical implementation, we adopt the discrete Markov chain model that is well suited for the analysis of financial high-frequency prices. The Markov chain framework facilitates simple expressions and elegant analytical results. The proposed estimator is consistent with a Gaussian limit distribution and we study its properties in simulations and an empirical application.

Markov chain, Filtering Contaminated Semimartingale, Quadratic Variation, Integrated Variance, Realized Variance, High Frequency Data

Abstract:
The Granger representation theorem states that a cointegrated vector autoregressive process can be decomposed into four components: a random walk, a stationary process, a deterministic part, and a term that depends on the initial values. In this paper, we present a new proof of the theorem. This proof enables us to derive closed-form expressions of all terms of the representation and allows a unified treatment of models with different deterministic specifications. The applicability of our results is illustrated by examples. For example, the closed-form expressions are useful for impulse response analyses and facilitate the analysis of cointegration models with structural changes.

Abstract:
We examine the impact of monetary policy on the S&P 500 using intraday data. The analysis shows an economically and statistically significant relationship between S&P 500 intraday returns and changes in the Fed funds target rate. The significance and magnitude of the response is dependent on whether the change was expected or unexpected. An expected change in the Fed funds target rate has no impact on prices in the broad equity market; however, an unexpected change of 25 basis points in the Fed funds target rate results in an approximate 48 basis points decline in the broad equity market's return. The speed of these market reactions is rapid with the equity market reaching a new equilibrium within 15 minutes.

Monetary policy, Exchange traded funds, High-frequency data