Invalid t-statistics

Transcription

1 A Demonstration of Time Varying Volatility & Invalid t-statistics Patrick M. Herb * this version: August 27, 2015 Abstract This paper demonstrates that using ordinary least squares (OLS) to estimate a model that exhibits time varying volatility can result in invalid t-statistics and a rejection of the true data generating process. Articial data is simulated for ten separate models that all have the same linear relationship but dierent volatility structures. Parameters are estimated for each model using both OLS with Huber- White heteroskedasticity consistent standard errors (robust standard errors) and an alternative model that compensates for the time varying volatility. In each case, OLS with robust standard errors rejects the true data generating process by more than the allowed p-value. This result indicates that the eciency gains from modeling volatility may reverse contemporary ndings in economics and nance. Keywords: time varying volatility, inecient estimator, invalid inference, biased variance estimator, consistency, OLS, GLS, WLS, FGLS, ARCH, GARCH * Herb: PhD Candidate, International Business School, Brandeis University, 415 South Street, Mailstop 32, Waltham, MA , pmherb@brandeis.edu,

2 1 Introduction It is well known that the OLS point estimator under heteroskedasticity (time varying volatility) is consistent and unbiased, but is not a minimum variance unbiased estimator; it is neither ecient nor asymptotically ecient. Additionally, the OLS variance estimator is generally biased and the covariance matrix may not be consistent. These undesirable eects of heteroskedasticity can result in unreliable inference; both condence intervals and t-statistics can be invalid. Left uncorrected or ignored, these eects can lead researchers to either accept an explanatory variable that is actually false or falsely reject an explanatory variable that is part of the true data generating process. This paper focuses on the latter. To highlight the gains in eciency that can be made by properly accounting for time varying volatility, ten dierent linear models are simulated such that they all have the same linear relationship between the dependent and independent variable. For each model, a dierent time varying volatility structure is constructed and implemented through the innovation at time t. The dependent variable is constructed twice: once where the innovation has constant variance and a second time where the magnitude of the constant variance innovation is altered by the time varying volatility. This approach makes known the true relationship between the dependent and independent variables, the true volatility structure for each model, and enables a comparison between the constant volatility process and the process exhibiting time varying volatility. Each model is estimated in three dierent ways. The rst estimation technique applies standard OLS to the constant variance model. This is a good bench mark for comparing the other models because even though the true model parameters are known, the random variability in the data will not yield perfect estimates. The second estimation technique applies OLS with Huber-White heteroskedasticity consistent standard errors (robust standard errors) to the model with time varying volatility. Invalid inference is demonstrated through the inaccuracy of t-statistics. Each model of time varying volatility falsely rejects 1

3 the true model. The third estimation technique applies either feasible generalized least squares (FGLS) with a known optimal weighting matrix or maximum likelihood (ML) for models with ARCH and GARCH volatility to the time varying volatility process. Every example presented results in an increase in the point estimate precision, a decrease in the variance (standard error) of the point estimate, an increase in the associated t- statistic and a probabilistic armation of the true data generating process. The results are then compared and the eciency gain between technique two and three is measured and reported. The structure of this paper is as follows. Section 2 describes the motivation and empirical methodology in greater detail. Section 3 presents the empirical results and volatility model specications. Section 4 provides a robustness evaluation using a monte carlo method, and section 5 provides some concluding remarks. All data are available for download at the author's website. 2 Empirical Methodology This section provides intuition for the eciency gains that can be made by using alternative estimation techniques that properly model and compensate for time varying volatility, explains how the data are constructed and simulated, and outlines the parameter estimation methodology. 2.1 Motivation The entirety of this paper works with a specic case of the linear regression model y = Xβ + e where y is a (T x 1) vector of observations, X is a (T x k) design matrix, β is a (k x 1) vector of coecients, and e is a (T x 1) random vector with mean zero E[e] = 0 and covariance matrix E[ee ] = Φ = σ 2 Ω, where Ω is a (T x T ) known positive denite 2

4 matrix and σ 2 is an unknown scalar. There are two cases for the matrix Ω: (1) when the variance is constant e t N(0, σ 2 ) and Ω = I T, and (2) with time varying volatility e t N(0, σt 2 ) and Ω I T. For both cases autocorrelation is set to zero so that E[e t e τ ] = 0 if t τ. Under constant variance, the best linear unbiased estimator for this model is the OLS estimator ˆβ = (X X) 1 X y. Under OLS, inference is achieved through the covariance matrix E[( ˆβ β)( ˆβ β)] = σ 2 (X X) 1 X ΩX(X X) 1, which reduces to σ 2 (X X) 1 when Ω = I T, and is estimated by ˆσ 2 (X X) 1, ˆσ 2 = e e T k, e = y Xβ. With time varying volatility, Ω I T and the OLS estimator remains unbiased and consistent but the covariance matrix is σ 2 (X X) 1 X ΩX(X X) 1, which is generally dierent than the covariance matrix under OLS σ 2 (X X) 1 σ 2 (X X) 1 X ΩX(X X) 1. This implies that the OLS estimator may no longer be the best linear unbiased estimator because it may no longer be a minimum variance estimator, and therefore not ecient. Furthermore, it can be shown that under time varying volatility the estimator ˆσ 2 = e e T k is generally a biased estimator of σ2. This results in incorrect standard errors, which invalids standard hypothesis tests. If the bias is positive, the acceptance regions will be wider and the probability of rejecting the null hypothesis will be lower than the correct one. Conversly, if the bias is negative, the acceptance regions will be smaller and the probability of rejecting the null hypothesis will be higher. 3

5 In economics and nance, most linear models are commonly estimated against the hypothesis that the coecient is not dierent than zero. Then, testing an independent variable's correlation with the dependent variable amounts to converting the estimated coecient into a t-statistic (t = ˆβ 0 σ β ) and comparing the t-statistic to Student's t- distribution. Large t-statistics that reject the null hypothesis in favor of the coecient diering from zero at some condence level (90%, 95%, 99%) are then reported by researchers and arguments are made as to why the variables are related. Variables that have smaller t-statistics than the accepted condence regions allow, can, sometimes opposite to theoretical presuppositions, be tossed aside as having little explanatory power. This paper demonstrates that a failure to correctly model and compensate for the volatility structure of a given dataset could result in the erroneous rejection of explanatory variables. Suppose data used to estimate a model has time varying volatility resulting in a positive bias. If the researcher relies on OLS and fails to correct the heteroskedasticity, she might falsely accept that the coecient is not dierent than zero. The positive bias can reduce the t-statistic to a level that fails to reject the null hypothesis. Also, not using a minimum variance estimator can result in inaccurate and inecient point estimates, large standard errors and small t-statistics. In other words, the researcher might have the correct model, but the conclusions reached through interpretation of the t-statistics can reject the correct model as a result of either the bias or not minimizing the variance of the estimator, or both. Using robust standard errors alone might result in a consistent estimate of the covaraince matrix, but might not be adequate to overcome the ineciency of the point estimator and the variance estimator. This paper demonstrates these eects and provides examples and data for researchers to experiment with. 4

6 2.2 Simulating Data For each model, an independent variable X (T x 1) and noise innovation e (T x 1) are generated such that X t N(0, 1), e t N(0, 1), T = Next, a specic time varying volatility structure is generated such that the variance at time t is a function of either time, lagged variance, lagged squared errors, or any combination of these so that σ 2 t = f(t, σ 2 t i, e 2 t j). Altering the innovation to impose the time varying volatility structure is straight forward e tv t = σ t e t e tv t N(0, σ 2 t ). With two dierent innovation vectors we can generate two dependent variables that have the same linear relationship and similar innovations in that they only dier by the variance through time. y = Xβ + e y tv = Xβ + e tv To ease interpretation of the tables that follow, the true parameter β was chosen the same for every model such that β = 0.5. The volatility structure for each model is reported in section 3 with it's corresponding tables and graphs. 5

7 2.3 Parameter Estimation As stated above, this paper uses two dierent estimation methods to reduce the eects of the imposed time varying volatility. The rst six models are estimated using feasible generalized least squares with a known optimal weighting matrix. This is possible because the optimal weighting matrix was chosen by the author and was used to create the data. The last four models have ARCH and GARCH volatility, and are estimated using maximum likelihood (ML). The maximum likelihood estimation simultaneously estimates the model parameters and volatility structure. A deeper understanding of both FGLS and ML can be found in just about any current graduate level econometrics text book. This section provides an illustration of only the most useful concepts of these techniques for the readers' intuition. White (1980) provided a parameter covariance matrix estimator which is consistent even when the disturbances of a linear regression model are heteroskedastic. This estimator has become familiar to most economists, and is often referred to as a robust estimator by econometric software packages. The estimator uses the OLS point estimator to calculate the disturbances of the regression and then uses the squared disturbances to estimate the time t variance. If ê t is the OLS generated disturbance at time t, then the estimated time t variance ˆσ 2 t is estimated by ê 2 t. Then, White's heteroskedasticity-consistent parameter covariance estimator is equal to (X X) 1 X ˆΩX(X X) 1 where ˆΩ = diag(ê 2 1, ê 2 2,..., ê 2 T ) or ˆΩ = diag((y 1 X 1 ˆβ) 2, (y 2 X 2 ˆβ) 2,..., (y T X T ˆβ) 2 ). White acknowledges potential eciency gains may be realized from a more careful modeling of the variance structure. The eciency gains from directly modeling the 6

8 volatility structure in this paper are two fold: (1) the point estimator shows improved precision and (2) the variance of the parameter covariance matrix estimator is reduced. Estimating ˆΩ using an extremely imprecise point estimate will result in imprecise variance estimates. Using robust standard errors does not improve the precision of the OLS point estimator. This paper uses a known volatility structure to highlight the eciency gains that can be made from correctly modeling the variance (volatility) structure. Generalized least squares with a known optimal weighting matrix is known as feasible generalized least squares. When the optimal weighting matrix is known, the FGLS estimator is a minimum variance unbiased estimator, and is superior to OLS. The rst step is to transform the linear model by multiplying it by a matrix P P y = P Xβ + P e, such that the matrix P has the property σ 2 P ΩP = σ 2 I T, and therefore Ω 1 = P P. Applying least squares to the transformed model results in the generalized least squares estimator ˆβ = (X Ω 1 X) 1 X Ω 1 y. The covariance matrix of the transformed model is σ 2 (X Ω 1 X) 1, which is dierent than the covariance matrix of the untransformed model σ 2 (X Ω 1 X) 1 σ 2 (X X) 1 X Ω 1 X(X X) 1. Finally, an unbiased estimator of σ 2 is given by 7

9 ˆσ 2 = (y Xβ) Ω 1 (y Xβ). T k Maximum likelihood is a computational estimation method that nds the parameters that maximize the probability of obtaining the sample actually observed. The likelihood function that is to be maximized typically begins as a product of probability distributions. Since it is typically easier to work in sums than products, the likelihood function is often converted to log form, and is called the log-likelihood function. If the values for {e t } are drawn from a normal distribution with zero mean and constant variance, then the likelihood of any realization of {e t } is 1 L t = ( 2πσ 2 )exp( e2 t 2σ ). 2 Now, if all of the realizations of {e t } are independent, the likelihood of the joint realizations of {e t } are the product L = T 1 ( 2πσ 2 )exp( e2 t 2σ ). 2 t=1 Taking the natural log of the above equation and remembering that e t = y t βx t gives the log-likelihood function for a linear model under the assumptions of OLS lnl = T 2 ln(2π) T 2 ln(σ2 ) 1 2σ 2 T (y t βx t ) 2. The above equation is possible because the conditional variance is the same as the unconditional variance for every realization of {e t }. For ARCH and GARCH models, the probability distribution must be modied to account for the conditional variance. Let h t be the conditional variance at time t. Keeping the same assumptions as above with the exception of constant variance, then the joint likelihood of the realizations {e t } are t=1 L = T 1 ( )exp( e2 t ), 2πht 2h t t=1 8

10 and the log-likelihood function is lnl = T 2 ln(2π) 1 2 T ln(h t ) 1 2 t=1 T t=1 ( e2 t h t ). In eect, this approach weights each observation of {e t } with its conditional variance. Standard errors are approximated numerically. 3 Empirical Results This section provides graphs of the depedent variable with time varying volatility for each time series, a detailed description of the volatility structure, and the estimation results for each model. In every case, the rst estimation results are of the constant variance model using OLS. The second estimation uses the same X and innovation as the rst, except the innovation has been modied by the time varying volatility as described in section 2. The second estimation uses OLS with Huber-White heteroskedasticity consistent standard errors (robust standard errors). In each case, OLS with robust standard errors rejects the true data generating process. The third estimation compensates for the time varying volatility either by FGLS with an optimal weighting matrix or maximum likelihood for the ARCH and GARCH processes. In each case, the variance compensated estimation results in the armation of the true data generating process. Lastely, the eciency gain is reported as the variance of the estimated coecient under OLS with robust standard errors divided by the variance of the estimated coecient from the variance compensated model. In every case there are eciency gains from the variance compensated estimation. All models use separate data. 3.1 Triangular Volatility This paper denes triangular volatility as volatility that is constantly increasing or decreasing over time. The graph of the dependent variable {y t } is triangular in shape. Model 9

11 1 is an example of increasing triangular volatility. The parameters used to simulate the volatility process used in this paper are σ t t T σ t =. 1 t = 1 This implies that volatility starts at 1 for the rst observation and increases by 0.05 at every time step. Table 1 Increasing Triangular Volatility Model 1 Variance Estimation Method ˆβ SE tstat pvalue σ 2 OLS (0.0325) σt 2 OLS, Robust SE (0.9415) σt 2 FGLS, Ω Known (0.2056) FGLS Eciency Gain: V ar( ˆβ OLS )/V ar( ˆβ F GLS ) = The rst line of table 1 reports that using OLS to estimate the constant variance model results in a fairly precise coecient point estimate, a small standard error, a large t-statistic and a high probability that the coecient is dierent from zero. Another way of saying this is that the independent variable X has explanatory power of the dependent variable y. This result would arm the true data generating process. The second line of the same table reports the estimation of model 1 with time varying volatility using OLS with robust standard errors. The results from this estimation technique on model 1 are an imprecise coecient estimate with the wrong sign, a large standard error and a small t-statistic implying a low probability that the coecient is dierent than zero. In other words, OLS with robust standard errors rejects the true data generating process. The third line of the same table reports the estimation of model 1 with time varying volatility using FGLS with an optimal weighting matrix. The eciency gain from correctly compensating for the time varying volatility results in a huge improvement in the point estimate precision, a reduction in the size of the associated standard error, and an increase in the magnitude of the t-statistic, which implies a high probability that the coecient is dierent than zero. In other words, correctly compensating for the 10

12 time varying volatility results in arming the true data generating process. Finally, the variance of the coecient estimate using OLS with robust standard errors is nearly 21 times as large as the variance of the coecient estimate using FGLS with an optimal weighting matrix. 150 Increasing Triangular Volatility 100 y t Decreasing Triangular Volatility y t Time Figure 1. Plot of the dependent variable {y t } exhibiting increasing and decreasing triangular volatility versus time. Similar results are demonstrated by decreasing triangular volatility (model 2). The parameters used to simulate model 2 are σ t t T σ t =. 1 t = T Here, volatility has a slope and the last observation has volatility equal to 1. Again, the rst line of table 2 is our benchmark. The second line shows that OLS with robust standard errors again rejects the true data generating process. The third line indicates that FGLS estimation with an optimal weighting matrix compensates for the time varying volatility and results in a more precise coecient estimate, a smaller standard error, and a larger t-statistic that probabilistically arms the true data generating process. The variance of the OLS estimator with robust standard errors is nearly 16 times as large as the FGLS estimator. 11

13 Table 2 Decreasing Triangular Volatility Model 2 Variance Estimation Method ˆβ SE tstat pvalue σ 2 OLS (0.0318) σt 2 OLS, Robust SE (0.8738) σt 2 FGLS, Ω Known (0.2186) FGLS Eciency Gain: V ar( ˆβ OLS )/V ar( ˆβ F GLS ) = Diamond & Buttery Volatility Diamond volatility increases for the rst half of the sample and then declines for the second half of the sample. The data generated for this paper starts with volatility equal to 1 and then increases linearly with slope 0.05 until the midway point or observation 500. Observation 501 then repeats observation 500 and then linearly declines with a slope of until it reaches a value of 1 on the last observation. The exact specications for model 3 are σ t t 500 σ t = σ t t T 1 t = 1, t = T. Table 3 Diamond Volatility Model 3 Variance Estimation Method ˆβ SE tstat pvalue σ 2 OLS (0.0303) σt 2 OLS, Robust SE (0.4331) σt 2 FGLS, Ω Known (0.1619) FGLS Eciency Gain: V ar( ˆβ OLS )/V ar( ˆβ F GLS ) = Buttery volatility starts out high and then declines linearly for the rst half of the sample, and then increases linearly for the second half. Similarly to diamond volatility, observations 500 and 501 are the same. The exact specications for model 4 are σ t t 500 σ t = σ t t T. 1 t = 500,

14 Diamond Volatility 50 y t Butterfly Volatility y t Time Figure 2. Plot of the dependent variable {y t } exhibiting diamond and buttery volatility versus time. Again, OLS estimation with robust standard errors of both diamond and buttery volatility result in rejecting the true data generating process. Properly weighting the errors by FGLS with an optimal weighting matrix increases eciency. The eciency gain results in an increase in the coecient point estimate precision, a reduction in standard error size, and an increase in the associated t-statistic implying an armation of the true data generating process. The variance of the OLS estimator with robust standard errors is times as large as the FGLS estimator for model 3, and times as large for model 4. Table 4 Buttery Volatility Model 4 Variance Estimation Method ˆβ SE tstat pvalue σ 2 OLS (0.0315) σt 2 OLS, Robust SE (0.4796) σt 2 FGLS, Ω Known (0.1691) FGLS Eciency Gain: V ar( ˆβ OLS )/V ar( ˆβ F GLS ) = Volatility Regime Switching Volatility regimes are periods of time when volatility is constant, but at certain points in time there is a discrete jump to another volatility state. There can be mutiple regimes 13

15 within a sample of observations. The key distinction for regime switching volatility is that the volatility stays constant throughout each regime or period of time. This paper gives an example of both a two regime and three regime sample. The two regime sample is dened as 5 1 t 500 σ t = t T The rst half of the data have constant volatility equal to 5. The second half of the data have constant volatility equal to 20. There is a discrete jump in volatility at observation 501. Table 5 Two Regime Volatility Model 5 Variance Estimation Method ˆβ SE tstat pvalue σ 2 OLS (0.0309) σt 2 OLS, Robust SE (0.4449) σt 2 FGLS, Ω Known (0.2106) FGLS Eciency Gain: V ar( ˆβ OLS )/V ar( ˆβ F GLS ) = For three volatility regimes, it might be intuitive to consider a time series that has a normal period, a period of crisis and a period of government intervention. The period of crisis may have higher volatility than the normal period, and a period of government intervention may have lower volatility than a normal period. With this in mind, this paper uses three volatility regimes such that 10 1 t 400 σ t = t t T For both volatility regime examples, OLS with robust standard errors reject the true data generating process. FGLS with an optimal weighting matrix increases the precision of the point estimate, reduces the size of the standard error, increases the size of the t- statistic and arms the true data generating process. The variance of the OLS estimator 14

16 Two Volatility Regimes y t y t Three Volatility Regimes Time Figure 3. Plot of the dependent variable {y t } exhibiting regime switching volatility versus time. with robust standard errors is times as large as the FGLS estimator for model 5, and 2.6 times as large for model 6. Table 6 Three Regime Triangular Volatility Model 6 Variance Estimation Method ˆβ SE tstat pvalue σ 2 OLS (0.0324) σt 2 OLS, Robust SE (0.3500) σt 2 FGLS, Ω Known (0.2169) FGLS Eciency Gain: V ar( ˆβ OLS )/V ar( ˆβ F GLS ) = ARCH Volatility Engle (1982) showed how to model the variance as an autoregressive process. Models with autoregressive conditional heteroskedasticity (ARCH) exhibit volatility persistence. Typical of autoregressive processes, future values depend on past realizations. Larger past realizations increase expected future realizations, while smaller realizations decrease expected future realizations. These models are very much related to the concept of volatility clustering, where volatility remains persistent for some time before changing. Most nancial data exhibit some form of volatility clustering; periods of low volatility followed by periods of high volatility. 15

17 Table 7 ARCH(1) Volatility Model 7 Variance Estimation Method ˆβ SE tstat pvalue σ 2 OLS (0.0315) σt 2 OLS, Robust SE (0.2405) σt 2 ML, ARCH(1) (0.1527) FGLS Eciency Gain: V ar( ˆβ OLS )/V ar( ˆβ F GLS ) = Model 7 has ARCH(1) volatility, which implies that the conditional variance h t depends on a constant and the lagged squared innovation e 2 t 1. Volatility is just the square root of the variance. The full specications for the example used in this paper are h t = α 0 + α 1 e 2 t 1 α 0 = 20 α 1 = Model 8 has ARCH(2) volatility. This model is similar to model 7 except that there is an additional lagged squared innovation e 2 t 2. The full model specications for the example used by this paper are h t = α 0 + α 1 e 2 t 1 + α 2 e 2 t 2 α 0 = 20 α 1 = 0.70 α 2 = Using OLS with robust standard errors to estimate these models results in rejecting the true data generating process for both model 7 and model 8. Interestingly, both ARCH processes are estimated using maximum likelihood with the correct number of ARCH terms and no additional information. Dierent than the FGLS estimation, no optimal weigthing matrix is supplied. Maximum likelihood estimates both the volatility 16

18 40 ARCH(1) 20 y t ARCH(2) y t Time Figure 4. Plot of the dependent variable {y t } exhibiting ARCH volatility versus time. parameters and the linear model parameters simulataneously. In both cases, estimation using maximum likelihood and weighting the errors with their conditional variance improves the coecients point estimate precision, reduces the size of the standard errors of the estimate, increases the magnitude of the t-statistics and arms the true data generating processes. The variance of the OLS estimator with robust standard errors is times as large as the ML estimator for model 7, and times as large for model 8. Table 8 ARCH(2) Volatility Model 8 Variance Estimation Method ˆβ SE tstat pvalue σ 2 OLS (0.0318) σt 2 OLS, Robust SE (0.4602) σt 2 ML, ARCH(1) (0.1801) FGLS Eciency Gain: V ar( ˆβ OLS )/V ar( ˆβ F GLS ) = GARCH Volatility Generalized autoregressive conditional heteroskedasticity (GARCH) is an extention of Engle's work by Bollerslev (1986) to model variance as an ARMA process. For GARCH processes, the conditional variance is related to both past innovations and past conditional variances (and a constant). Model 9 is a GARCH(1,1) process, which implies that it has 17

19 one lagged innovation and one lagged conditional variance. The example used in this paper is fully specied by h t = α 0 + α 1 e 2 t 1 + b 1 h t 1 α 0 = 3 α 1 = 0.20 b 1 = Table 9 GARCH(1,1) Volatility Model 9 Variance Estimation Method ˆβ SE tstat pvalue σ 2 OLS (0.0300) σt 2 OLS, Robust SE (0.2015) σt 2 ML, GARCH(1,1) (0.1771) FGLS Eciency Gain: V ar( ˆβ OLS )/V ar( ˆβ F GLS ) = Model 10 is a GARCH(2,2) process, which implies that it has two lagged innovations and two lagged conditional variances. The full specication of the example used in this paper is h t = α 0 + α 1 e 2 t 1 + α 2 e 2 t 2 + b 1 h 2 t 1 + b 2 h 2 t 2 α 0 = 3 α 1 = 0.05 b 1 = 0.65 b 2 = The results reported in the corresponding tables echo those reported for the other models. In both cases, estimation using OLS with robust standard errors rejects the true data generating process. Modeling the volatility and weighting the errors with 18

20 GARCH(1,1) 20 y t GARCH(2,2) 20 y t Time Figure 5. Plot of the dependent {y t } variable exhibiting GARCH volatility versus time. their conditional variance results in maximum likelihood estimation that improves the coecients point estimate precision, reduces the standard errors, increases the t-statistics and arms the true data generating processes. The variance of the OLS estimator with robust standard errors is times as large as the ML estimator for model 9, and times as large for model 10. Table 10 GARCH(2,2) Volatility Model 10 Variance Estimation Method ˆβ SE tstat pvalue σ 2 OLS (0.0326) σt 2 OLS, Robust SE (0.2711) σt 2 ML, GARCH(2,2) (0.1822) FGLS Eciency Gain: V ar( ˆβ OLS )/V ar( ˆβ F GLS ) = Monte Carlo Robustness The results for any random process will vary depending on both parameter choice and the particular draw of random variables. The last section demonstrated that given a particular set of parameters and volatility process, estimation using OLS with robust standard errors could reject the true data generating process. This section seeks to answer how often this might happen given a set of particular parameter values and volatility 19

21 process. To answer this question, a monte carlo method is employed. Table 11 Monte Carlo Results < < > < p>0.10 p<0.10 Result Model ˆβ 0.5 SE tstat pvalue OLS FGLS/ML Reversal Monte carlo results for 100,000 simulations. Values represent number of occurences divided by number of simulations. The monte carlo method seeks to quantify the frequency that the previous section results would occur, given the same parameters and volatility process. In particular, it would be of interest to know how often parameter precision is improved, standard errors are reduced, t-statistics are increased and p-values are decreased. Additionally, it may add value to know how often OLS with robust standard errors falsely rejects the true data generating process and how often the volatility compensated model arms the true data generating process. Finally, the monte carlo method also gives us the frequency that the results are reversed; when OLS with robust standard errors rejects but the variance compensated estimation arms the true data generating process. For each model, the process is repeated 100,000 times using the same parameter values and volatility process as described in sections 2 and 3. To measure the precision of the estimate, the distance from the true parameter value is measured as distance = ˆβ 0.5. If the distance from the parameter estimate to the true value is reduced by the variance compenated estimation, the precision is said to have improved. Comparing standard errors is straight forward; the standard errors should generally be smaller after compensating for the variance. Since it is known that the true parameter value is a positive 20

22 number, larger t-statistics can be thought of as an improvement. Moving inversely to t-statistics, smaller p-values can also be considered an improvement. There is also the strange case when a t-statistic can increase but the associated p-value gets smaller. This can occur when at least one t-statistic (estimated coecient) is negative. Comparing how often each technique falsley rejects the true data generating process could use some illustration. The null hypothesis for each estimation technique is that the estimated parameter is not dierent from zero. A critical p-value of 10% is chosen ex-ante. If the ex-post p-value is smaller than 10%, then the inference statement would conclude with 90% condence that the estimated parameter is dierent than zero, while a larger than 10% p-value would conclude that the estimated parameter is not dierent than zero. Since the null hypothesis is false, a p-value that is larger than 10% would result in the failure to reject the false null hypothesis, which is known as a type II error. A type II error results in a rejection of the true data generating process as it falsely implies that the estimated coecient does not explain the dependent variable. Therefore, to compare the two estimation techinques, it may be useful to compare how often OLS with robust standard errors rejects the true data generating process and how often the variance compensated model arms the true model. Additionally, it might be of interest to see how often a result is overturned; where OLS with robust standard errors rejects the true model and the variance compensated estimation arms the true model. The results for each simulation are then stored and a percentage is calculated for each category. The percentage represents how often the variance compensated estimation technique results in an improvement similar to the results outlined in section 3. For example, looking at table 11, 85.26% of the simulations resulted in improved precision for model 1's coecient estimate, which is measured as the distance from the actual value. Similarly, 99.45% of the simulations reduced the size of the standard error for model 9. Model 6 experienced an increase in the associated t-statistic size for 83.65% of the simulations. The size of the p-value was decreased for 91.91% of the simulations for model 7. Estimation using OLS with robust standard errors resulted in rejecting the true 21

23 data generating process for 70.59% of the simulations for model 5. Estimation of model 7 using maximum likelihood resulted in arming the true data generating process for 93.19% of the simulations. Finally, correctly modeling the volatility structure overturned the results for 65.52% of the simulations of model 3. 5 Summary While the OLS estimator remains unbiased and consistent with time varying volatility, it's ineciency may cause researchers to falsely reject the true data generating process. Properly modeling the volatility of a given time series could dramatically alter researcher outcomes. This paper provides ten examples of time series that falsely reject the true data generating process when estimated by OLS with robust standard errors. Additionally, the true data generating process may be armed by alternative estimation techniques that properly weight the errors of a regression. Relying on OLS with robust standard errors alone may or may not be the best approach for researchers. 22

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