Preliminary and Incomplete – Do not Quote

Autocorrelation-Corrected Standard Errors Using Moment Ratio Estimates of the Autoregressive/Unit Root Parameter

J. Huston McCulloch

Ohio State University

May 5, 2009

The author thanks participants at the Ohio State University Econometrics Seminar, for helpful comments and suggestions. All remaining errors and omissions remain the responsibility of the author. URL to PDF of latest version of paper, with color graphs: http://www.econ.ohio-state.edu/jhm/papers/MomentRatioEstimator.pdf Keywords: Method of Moments, Autoregressive processes, Unit Root Processes, Regression Errors, Consistent Covariance matrix, GDP trend growth, Demand for monetary base JEL codes: C13 (Estimation), C22 (Time Series Models), E41 (Demand for Money)

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ABSTRACT

A Moment Ratio Estimator is proposed for the parameters of an Autoregressive (AR) model of the error in an Ordinary Least Squares (OLS) linear regression. Although it is computed from the conventional residual autocorrelation coefficients, it greatly reduces their bias, and provides corrected standard errors with far less bias than alternatives. The estimator is in the spirit of the Median Unbiased estimator of Andrews (1993) and McCulloch (2008), but is more easily computed and provides smaller standard error bias in most cases. The presence of a unit root in the errors, and therefore the absence of a cointegrating relationship, requires reposing the problem, but does not by itself indicate that an OLS correlation between the variables is spurious. Hypothesis testing is standard, provided it is based on squared quasi-differenced residuals, and not on the squared residuals themselves.

Although the present paper is restricted to the AR(1) case, the approach is readily extendable to higher-order AR processes. An exact unit root test similar to that of Andrews (1993) is implemented for the AR(1) case. The Moment Ratio estimator is applied to an income trend line regression, as well as to a monetary base demand function. In both cases, the Moment Ratio autoregressive coefficient estimate is quite close to unity, and a unit root in the errors cannot be rejected. However, the trend slope remains highly significant in the income trend line regression, and both the income elasticity and interest semi-elasticity remain highly significant in the base demand equation, even when a unit root is imposed. It is observed that despite their consistency, the popular HAC standard errors of Newey and West (1987) can greatly overstate the precision of OLS coefficient estimates with sample sizes and serial correlation commonly found in economic studies when, as has become standard, “automatic bandwidth selection” is employed.

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1. Introduction Serial Correlation is a pervasive problem in time series models in econometrics, as well as in statistics in general. When, as is often the case, positive serial correlation is present in both the errors and the regressors, it has long been well known that Ordinary Least Squares (OLS) standard errors are generally too small, and hence the derived t-statistics too large.1 If the form and parameters of the error serial correlation were known, it would be straightforward to compute correct standard errors for OLS regression coefficients. However, observed regression residuals are typically much less persistent than the unobserved regression errors. Correlations estimated directly from the regression residuals therefore provide inadequate indication of the serial correlation that is actually present. This problem is particularly severe as the persistence in the errors approaches or even reaches a unit root. The present paper proposes a Moment Ratio (MR) estimator for the parameters of an Autoregressive (AR) model of the errors in an OLS regression. Although it is computed from conventional correlation coefficients, it removes their negative bias, and provides standard errors with far less bias than alternatives. The estimator is in the spirit of the Median Unbiased estimator of Andrews (1993) and McCulloch (2008), but does not require laborious Monte Carlo simulation of the distribution of the sample autocorrelations, and actually provides less biased standard errors in most cases. The non-stationary unit root case greatly increases the variance of OLS coefficients and requires reposing the problem, but otherwise presents no insurmountable difficulties. In particular, the presence of a unit root in the errors does not by itself indicate that the OLS correlation between two variables is spurious, provided the regressors include a trend or trending variable(s). Hypothesis testing is then standard, provided it is based on the sum of squared quasi-differenced residuals, and not on the sum of squared residuals itself.

An exact unit root test similar to that of Andrews (1993) is developed. However, it is found that it is not ordinarily beneficial to impose a unit root when one cannot be rejected, unless the estimated autoregressive coefficient is quite close to unity. The Moment Ratio estimator is applied to an income trend-line regression, and also to a monetary base demand function. In both cases, the MR standard errors are much higher than the alternatives. However, the OLS trend slope remains highly significant in the income trend line regression, and both the income elasticity and interest semi-elasticity remain highly significant in the base demand equation after correction for the serial correlation. In both cases the estimated autoregressive coefficient is quite close to unity, but still just under the value for which it is preferable to impose a unit root.

1 This understanding goes back at least to Bartlett (1935) and Quenouille (1952).

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Nevertheless, the slope coefficients all remain significant even when a unit root is imposed. It is also observed that despite their consistency, the HAC standard errors of Newey and West (1987) can greatly overstate the precision of OLS coefficient estimates with sample sizes and serial correlation commonly found in economic studies when, as has become standard, “automatic bandwidth selection” is employed. Practitioners are always in search of big t-statistics and therefore small standard errors, so it is entirely understandable that the HAC under-correction for serial correlation has become so popular. However, it is bad econometric practice to systematically overstate the significance of one’s results by deliberately choosing a deficient estimator. Section 2 below develops the MR estimator in the case of stationary AR(1) errors, and compares the derived standard errors to conventional alternatives. Section 3 considers the non-stationary unit root case. Section 4 investigates the bias in MR standard errors using Monte Carlo simulations, and develops an exact unit root test. Section 5 outlines extension of the AR(1) MR estimator to a more general AR(p) process, but leaves this extension to future research. Section 6 applies the MR estimator to a regression of real income on a time trend, and Section 7 applies it to a real monetary base demand equation. Section 8 concludes. 2. Stationary AR(1) errors Consider a time-series linear regression of the form (1) εXβy +=where X is an n × k matrix of exogenous regressors whose first column is ordinarily a vector of units. We assume that the n × 1 error vector ε has mean 0, is independent of X, and, if stationary, has a time-invariant autocovariation structure, ( )||)'E( ji−== γεεΓ . The OLS estimator of β, εXXXβyXXXβ ')'(')'(ˆ 11 −− +==then has covariance matrix2

. (2) ( )

,)'(')'()'(')'(

)'('')'(E)ˆ(

110

11

11

−−

−−

−−

=

=

=

=

XXXRXXXXXΓXXXX

XXXεεXXXβCovC

γin terms of the population autocorrelation matrix ( ) 0|| / γρ ΓR == − ji . (3)

The vector of observed OLS residuals equals the “annihilator matrix” M times the vector of unobserved errors: 2 See, e.g., Greene (2003: 193).

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. (4) MεεXXXXIe =−= − )')'(( 1

Define ,1,,0),(tr −== njs jj Kee'

where the j-th order trace operator trj( ) for an n×n matrix A = (ai,j) is defined by

. ∑−

=+=

jn

ijiij a

1,)(tr A

The sample autocorrelations are then customarily3 computed from the residuals as 1,,0,/ 0 −== njssr jj K . (5) In general, ,)'E(E MΓMMεεMee' == (6) so that )(tr)(trE 0 MRMMΓM jjjs γ== . (7)

Under the classic OLS assumption IΓ 0γ= , (3) becomes . (8) 1

0 )'( −= XXC γOLS

In this case,

kn

ss

−= 02

is an unbiased estimator of γ0. Furthermore, (9) 12 )'(ˆ −= XXC sOLS

is an unbiased estimator of C. However, when, as is often the case, the errors and regressor(s) are both positively serially correlated, s2 is no longer unbiased and will underestimate the variances of the .

OLSC

jβ If Γ, or even R, were known, Generalized Least Squares (GLS) would provide the efficient estimator of β, along with an unbiased estimate of its variance.4 However, when the covariance structure must be computed from the regression residuals, Hayashi (2000: 59) warns that the finite-sample properties of the Feasible GLS estimator of β are unknown. The present paper therefore restricts itself to the problem of estimating the covariance matrix of the OLS estimator, even in the unit-root and near-unit-root case. 3 This definition follows Hayashi (2000: 408), Greene (2003: 268), and others, by taking the ratio of the sum of n-j terms to that of n terms, to obtain what might be called the weak sample autocovariances. The strong autocovariance, defined as the ratio of the average value of etet-j to the average value of ej

2, by analogy to the population autocovariance γj, is larger by a factor of n/(n-j) and arguably preferable, but the convention employed does not affect the proposed Moment Ratio estimator of the φj, provided it is used consistently. 4 Choi, Hu and Ogaki (2008) thus recommend estimating regressions with unit root and near-unit-root errors more efficiently by Feasible GLS (FGLS). However, they still estimate the AR(1) coefficient by an OLS regression of the OLS residuals on one lag of themselves and therefore employ a biased (though consistent) estimate of this coefficient. Basing FGLS on the less-biased MR estimate of the AR(1) coefficient instead would yield superior finite sample properties.

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The serial correlation in econometric time series regressions is often approximately AR(1) in structure:

,1 ttt u+= −εφε (10) where |φ| ≤ 1 and the innovations ut are iid. Under (10), the population autocorrelations are . (11) j

j φρ =

The unit root case φ = 1 requires reposing the problem somewhat, and is discussed in the next section. For the moment, we therefore assume φ < 1. Figure 1 illustrates , (12) )/()(tr/E 00

2 kns −= MRMγthe expectation of s2 relative to the true variance γ0 of the regression errors, as a function of the true AR(1) parameter φ, in the case of the simple trend line model, tt ty εββ ++= 21 , (113) with sample size n = 100. For φ > 0 this ratio is less than unity, and hence s2 is downward biased. This bias depends on the observed regressor matrix X, by way of the “annihilator matrix” M, but unfortunately, it is also a function of the unknown parameter φ, via the correlation matrix R.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

Bias in s2 (trend line model, n = 100)

true φ

Es2 /

γ 0

Figure 1.

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Bias in s2 relative to the true variance γ0 of the regression errors, in a trend line regression with AR(1) errors, as a function of the true AR(1) parameter φ.

We define the Moment Ratio function for r1 as the ratio of the population moments whose sample counterparts define r1 per (5) and (7):

)(/tr)(tr);ψ( 01 MRMMRMX =φ . (14) The value of ψ( ) depends on φ through the correlation matrix R, and also on X via M. Fortunately, however, it does not depend on the unknown coefficient vector β. Figure 2 illustrates this value as a function of φ, as in Figure 1 for the special case of a trend line regression with n = 100. A 45 degree line representing the true value of φ is also plotted.

0 0.2 0.4 0.6 0.8 10

0.1

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0.5

0.6

0.7

0.8

0.9

1Moment Ratio Bias in r1 (trend line model, n = 100)

true φ

ψ( φ

,X) =

Es 1/E

s 0

ψ(φ,X)45 deg

Figure 2

It may be seen that r1 already has a small downward bias (in the Moment Ratio sense) when φ = 0, and that this downward bias increases as φ increases to 1. The bias may also be computed for φ < 0, but it vanishes as φ ↓ -1. Hence, only the more commonly encountered case φ ≥ 0 is illustrated.

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The present paper proposes to eliminate most of this bias in r1 as an estimator of φ, simply by numerically evaluating the inverse of the function );ψ( Xφ , with respect to its first argument, at the empirical r1: . (15) );(ψˆ

11 XrMR −=φ

This inverse function is illustrated in Figure 3 below.

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Moment Ratio estimator of φ (trend line model, n = 100)

r1 = s1/s0

φhat

MR

ψ-1(r1,X)

45 deg

Figure 3

Once has been found, the correlation matrix R may be estimated by

);(ψˆ1

1 XrMR −=φ

( )||)ˆ(ˆ jiMRMR −= φR . (16) A natural estimator of γ0, that is unbiased by (7), would be . (17) )ˆ(tr/ 00 MRM MRsUnfortunately, however, it does not lead to a consistent estimate of the variance of the innovations in the limit as φ approaches unity, as discussed in the next section. Nevertheless, this problem can be avoided by instead basing the estimator of γ0 on the quasi-differenced residuals: , eQd φφ =

where the quasi-differencing operator Qφ = (qij) is defined by qi,i+1 = 1, qi,i = -φ, and qi,j = 0 otherwise. We then have

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φφφφ QMRMQdd ′=′E , (18) so that

)ˆ(tr/ˆ ˆˆ0ˆˆ0 MRMRMRMRMRMR

φφφφγ QMRMQdd ′′=

is a natural and at least approximately unbiased estimator of the variance of the errors. The Moment Ratio estimator of the covariance matrix C of as given in (2) is then OLSβ . (19) 11

0 )'(ˆ')'(ˆˆ −−= XXXRXXXC MRMRMR γIf desired, the variance of the AR(1) innovations may then be estimated by ( )2

02 )ˆ(1ˆˆ MRMRu φγσ −= (20)

Andrews (1993) has noted that a pattern similar to that in Figure 2 emerges when the median of the Monte Carlo distribution of the OLS estimator of φ is plotted against φ . Andrews proposes that an Exactly Median Unbiased estimator of φ be computed by numerically inverting this median function. McCulloch (2008) implements and extends Andrews’ method to find exactly median unbiased estimators of all the coefficients of an AR(p) model for regression errors, and uses this to compute a Median Unbiased, Autocorrelation Consistent Covariance matrix for regression coefficients. However, the Moment Ratio approach of the present paper gives similar results, without the tedious and noisy Monte Carlo simulation at each step of the numerical search. Figure 4 below illustrates the root mean square bias in several estimators of the standard error of the slope coefficient in a trend line regression with n = 100.5 While the OLS standard error computed from (9) (bottom, magenta line) is right on the money when φ = 0, its RMS is less than 10% of the true value as φ reaches 1. The red line (third from bottom) depicts the RMS bias of the standard AR(1) standard error, computed from (2) with s2 in place of γ0, and with R estimated directly from r1. This is generally a big improvement, but has a small downward bias even for small φ, that arises from the small bias in r1 when φ = 0. The s.e. bias exceeds 60% of the true value as φ approaches unity. The green line (fourth from the bottom) replaces s2 with the value that would be unbiased if r1 were the true φ according to (12), but still estimates R directly from r1. This is an improvement over the standard AR(1) standard error, but still suffers from the bias in r1. The top (blue) line depicts the true standard error, i.e. the true standard deviation of , as computed from (2) with the true φ and γ

2β0.

5 For ease of reading a black-and-white printout of this paper, the legend in this and other figures lists the series in order from top to bottom.

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

SE of trend slope, relative to true φ, γ0 (n = 100)

true φ

SE

of t

rend

slo

pe, r

elat

ive

to tr

ue φ

, γ0

true SEAR(r1, γ0(r1))

AR(r1, s2)

NW HAC bw4OLS

Figure 4

The truncated-kernel Heteroskedasticity and Autocorrelation Consistent (HAC) covariance matrix, introduced by Newey and West (1987), is now widely used by economists to “correct” the standard errors of OLS time series coefficients for serial correlation. Greene (2003: 201) reports that its use is now “standard in the econometrics literature.” Hayashi (2000: 409-12) mentions only it, the similar Quadratic Spectral HAC of Andrews and Monahan (1992), and the Vector Autoregression HAC (VAR-HAC) method of den Haan and Levin (1996) as appropriate methods for correcting OLS standard errors for serial correlation. Stock and Watson (2007) present HAC as the only method worthy of consideration. The Newey-West truncated-kernel HAC estimator of C is , 11 )'(')'(ˆ −−= XXFXXXXCHAC

where ( )|)(| jikee ji −=F and (21) )0,/)max(()k( mjmj −=is the truncated Bartlett Kernel function for some bandwidth m. Most econometric packages provide “automatic bandwidth selection” for HAC, using a formula similar to the following “benchmark rule” recommended by Stock and Watson (2007: 607): ⎣ ⎦5.075.0 3/1 += nm , (22)

11

which yields m = 4 for n = 100.6

In the benchmark case of homoskedasticity considered in the present paper, the expectation of the HAC estimator is , (23) 11 )'(')'(ˆE −−= XXXHXXXC HACHAC

where ( ) ( ) MΓMHH ==−= jiji

HAC hjikh ,, and,|)(| . HAC thus effectively employs only the first m-1 sample autocovariances, and replaces the others with zeros. At the same time it down-weights the autocovariances it does not discard by the Bartlett kernel factor (m-j)/m. It also uses the regression residuals as if they were the errors themselves. For all three of these reasons, it tends to underestimate the coefficient variance for a regressor which is itself serially correlated. However, the amount by which it does this depends on both the ρj and the degree of serial correlation of the regressors themselves. The cyan line in Figure 4 above (second from the bottom) depicts the RMS bias in the HAC standard error for the same regression as the other lines, using bandwidth m = 4. Although it provides some improvement over the OLS standard errors, the already seriously deficient AR(1) standard errors are far superior. See McCulloch (2008) for further discussion of the HAC standard errors. 3. The Unit Root Case φ = 1 The unit root case φ = 1 poses no insurmountable problems, so long as there is a constant term and a trend or trending variable(s) in the original regression (1), and so long as the quasi-differenced residuals are used to estimate the variance of the innovations. In particular, it does not in itself indicate a “spurious regression” in which ordinary tests for significance of the relationship break down. It does, however, require reposing the problem, so as to replace certain undefined mathematical expressions with their limiting values. As is well known, the OLS coefficient estimates are inconsistent when both the regressor(s) and the errors contain a unit root with no drift. See e.g. Choi, Hu and Ogaki (2008: 330). Conditional on the observed regressor(s) and the finite innovation variance, the limiting coefficient error is Gaussian. However, there may be a problem of estimating the innovation variance from even the differenced OLS residuals if the residuals themselves are not consistently estimated.

Nevertheless, when the regression includes a constant and a time trend, the slope coefficient becomes √n consistent: Hayashi (2000: 570), e.g., shows that after scaling the horizontal axis by n and the vertical axis by √n, the slope coefficient is Gaussian about its

6 EViews, following a suggestion of Newey and West, uses ⎣ ⎦ 1)100/(4 9/2 += nm , which has very similar effect for n in the range 50 – 2000, and generally identical effect in the range 400 – 1000. The notation ⎣x⎦ indicates the floor function of x.

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true value with a finite variance. Without the scaling, the slope coefficient is therefore Gaussian with standard error proportional to 1/√n. Likewise, if the regressor is unit root with non-zero drift, the drift will dominate the unit root noise for large n, and so the regressor will act as if it were a time trend. This section is therefore limited to the case in which the regressors include either a time trend or a trending variable. As φ ↑ 1, each element of the unconditional covariance matrix Γ becomes infinite, holding the variance of the innovations in (10) constant. Furthermore, each element of the correlation matrix R becomes unity in the limit. These matrices are therefore no longer useful or informative, and the problem must be reposed without them.

2uσ

Although a random walk has infinite unconditional variance and covariances, its variances and covariances are all well defined conditional on its value at any point in time, say t = 0. Furthermore, as long as there is a constant term in the regression, the OLS residuals will sum to zero regardless of the value of the random walk at t = 0, so that the actual value of ε0 does not matter for their properties. The conditional covariance matrix of the errors, taken conditional on ε0, is , 0

200 )|'E( WεεΓ uσε ==

where ( )),min(0 ji=W . Equation (6) then becomes ( ) MMWMΓMee'ee' 0

200|EE uσε ===

so that )(trE 0

2 MWMjujs σ= . (24) The limiting value of );ψ( Xφ , as plotted in Figure 2, then becomes . (25) )

)

(/tr)(tr);1ψ( 0001 MWMMWMX =Due to sampling error, the actual value of r1 could be above or below this value. When inverting this function as in (15) and Figure 3, any value of r1 above should simply be identified with .

);1ψ( X1ˆ =MRφ

In the unit root case, the variance of the errors is infinite, and hence uninformative. However, the variance of the innovations remains informative, and enables us to estimate the covariance of the regression coefficients, conditional on the arbitrarily chosen reference point ε

2uσ

0, as . (26) 1

012

0 )'(')'(ˆˆ −−= XXXWXXXC uMR σ

The innovation variance could be estimated without bias, using (24), by 2

uσ . (tr/ 000 MWMsBut unfortunately, this unbiased estimator, which is the limiting value of (17) taken together with (20), is not consistent, even in a trendline regression (see, e.g. Hayashi 2000:570-71). Simulations with Gaussian errors indicate that its distribution is

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approximately a scaled χ2 with 2.5 degrees of freedom, using either n = 100 or n = 1000, for a model in which only a constant term is estimated.7 In a linear trend model, it is approximately scaled χ2 with 5 degrees of freedom, using either n = 100 or n = 1000.

However, using the quasi-differenced residuals (which are in fact simply first-differenced in the unit root case being discussed here), equation (18) becomes

1012

10101111 )|E(E QMWMQQMΓMQdddd ′=′=′=′ uσε , whence

)(E 1012

11 QMWMQdd ′=′ truσ , so that )(/ˆ 10111

2 QMWMQdd ′′= truσ (27) is an unbiased estimator. Furthermore, simulations with Gaussian errors indicate that the distribution of this estimator is approximately scaled χ2 with n-k degrees of freedom, for both the mean and trend line models, and for both n = 100 and 1000. Although there is no guarantee that it is independent of , inference on the coefficients with t or F statistics based on (26) and (27) should be at least approximately valid.

ββ −ˆ

If a reference point other than ε0, say εt, is chosen, the conditional covariance of the errors becomes , tutt WεεΓ 2)|'E( σε ==where ( )t

jit w ,=W , with

⎩⎨⎧ <<>>−−

=.,0

)()(|),||,min(|, else

tjandtiortjandtitjtiwt

ji

Since MWtM does not depend on the choice of t, exactly the same values of and will be obtained. Furthermore, except for its first row and column,

);1ψ( X2ˆ uσ

112 )'(')'(ˆˆ −−= XXXWXXXC tuMRt σ

does not depend on the choice of t. The estimated standard error of the constant term β1 therefore does depend on the reference point defined by the choice of t, while the slope coefficients β2, ... βk do not. In particular, the regression F statistic for the joint hypothesis β2 = 0, ... βk = 0 is at least approximately valid despite the unit root in the errors, and is invariant to the arbitrarily chosen reference point. If the one-step-ahead forecast of yn+1 (conditional on the time t = n+1 values of the regressors) is of particular interest, the reference point t = n+1 may be a convenient choice convenient. In this case, ( ))1,1min(1 jninn −+−+=+W .

7 Although s0 can be expressed as a linear combination of squared N(0,1) random variables, the weights are unequal so that its distribution is not exactly χ2. Nevertheless, the relation between the distribution’s simulated mean and variance is the same as for the χ2 with the indicated degrees of freedom.

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Another particularly natural reference point, one that does not single out any particular point in the sample period, is the mean error n'ε /ει= , where ι = (1, ... 1)’. Conditioning on this value, we have 112 )'(')'(ˆˆ −−= XXXWXXXC εε σ u

MR , where ZWZW 0=ε and (28) nn /'ιιIZ −= Since the unconditional variance of the errors is infinite in the unit root case, the unconditional variance of is also infinite, as is its unconditional covariance with the other coefficients. In the unit root case, we may therefore write

OLS1β

, NCNC MRMR0

ˆˆ =where N is the n×n identity matrix In, with its (1,1) element replaced by ∞. 4. Monte Carlo Properties of Moment Ratio Estimator and Unit Root Test Figure 5 below shows the Moment Ratio Function ψ(φ, X), along with the Monte Carlo mean, median, quartiles, and 5th and 95th percentiles of the distribution of r1, as a function of the true value of φ, again for the trend line model with n = 100, under the additional assumption of Gaussian errors.8 The median line is essentially the same as the median function inverted by Andrews (1993) to determine his Median Unbiased estimator of the autoregressive parameter φ.9 The Moment Ratio Function is indistinguishable from the median for values of φ near 0, but rises somewhat above the median as φ increases. Nevertheless, it remains well within the quartiles, and at least approximately corrects the median bias of r1 when inverted. The downward skewness of the distribution of r1 for φ < 1 pulls the mean of the distribution down below the median by a comparable amount.

8 This simulation was performed with m = 99,999 replications using MATLAB’s randn(‘state’) Gaussian random number generator, for φ in steps of 0.01. The same seed was used for each value of φ in order to make each simulated percentile a smooth function of φ. 9 Andrews in fact bases his estimator on an OLS regression of yt on yt-1, a constant, and t, rather than on the first order autocorrelation of the residuals of a regression of yt on a constant and t. The two approaches are nevertheless very similar. As demonstrated in McCulloch (2008), the latter approach is easily applied to the residuals of a general OLS regression with higher-order autoregressive errors.

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.2

0

0.2

0.4

0.6

0.8

1Moment Ratio function with distribution of r1 (trend line, n = 100)

φ

Mon

te C

arlo

dis

tribu

tion

of r 1 =

s1/s

0, ψ( φ

,X)=

Es 1/E

s 0

95%75%

ψ(φ,X)medianmean25%5%

Figure 5 Figure 6 below shows the probability, from this simulation, that r1 will lie below the true value of φ, the Moment Ratio function, the median of r1 (just for reference), and the mean of r1, respectively, from top to bottom. The first schedule rises quickly and becomes indistinguishable from unity by φ = 0.96. (The true simulated probability there is 0.9943). The quantile of r1 that the Moment Ratio function corresponds to starts at 0.50 for φ = 0, but rises to 0.6162 at φ = 1. The quantile corresponding to the mean declines from 0.50 at φ = 0 to 0.4216 at φ = 1.

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.4

0.5

0.6

0.7

0.8

0.9

1Quantiles of distribution of r1 (trend line, n = 100)

φ

Mon

te C

arlo

pro

babi

lity

r 1 less

than

indi

cate

d va

lue

true φMR functionmedianmean

Figure 6

Because the Moment Ratio function lies above the median in Figure 5 for φ > 0, and both are increasing functions of φ, is necessarily less than the Median Unbiased estimator computed by inverting the Monte Carlo median function at r

MRφ1, and therefore

provides a less conservative adjustment for serial correlation, at least for the problem simulated.

The green (central) line in Figure 7 below shows the Monte Carlo RMS of the

Moment Ratio standard error for the OLS trend slope coefficient, relative to the coefficient’s true standard deviation represented by the red (bottom) line, for comparison to Figure 4 above. It may be seen that the Moment Ratio estimator completely eliminates the downward bias in the alternative standard error estimators for values of φ < 0.99, and in fact is somewhat too conservative in this range. The upward bulge in this curve for φ in the approximate range (0.8, 0.98) is caused by the increasing probability of falsely obtaining = 1, in conjunction with the convexity of the standard error as a function of φ.

MRφ

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.5

1

1.5

2

2.5

3

3.5

4

4.5Monte Carlo RMS SE of trend slope, rel. to true SE (n = 100)

true φ

RM

S S

E o

f tre

nd s

lope

, rel

. to

true

SE

Imposing Unit Root if can't reject@5%

MR φhat throughoutTrue SE

Figure 7

Above approximately 0.979, the Moment Ratio RMS standard error in Figure 7

falls below unity, relative to the true standard error, reaching 0.698 at φ = 1. Some downward bias is inevitable at this boundary, since sampling error can only produce too low a standard error. However, even this 30.2% maximal downward bias is quite small in comparison to that of the alternatives illustrated in Figure 4.

Since, as implied by Figures 5 and 6, standard errors based on the Median

Unbiased estimator of φ are even more conservative than the Moment Ratio standard errors, the Median Unbiased approach will only aggravate the small upward bias already present in the Moment Ratio estimates for most values of φ. The Moment Ratio estimator therefore dominates the Median Unbiased estimator, in terms of both its ease of computation and its standard error bias. The Monte Carlo distribution of r1 for φ = 1, illustrated at the right edge of Figure 5 above, provides a simple test for a unit root, under the assumption of Gaussian errors, that is exact to within Monte Carlo sampling error: If r1 is less than say the 5th percentile of this distribution (0.7777 for the illustrated trend line regression with n = 100), a unit root can be rejected with a 5% test size. The only practical way to perform this test is with a Monte Carlo simulation comparable to that required for Andrews’ (1993) Median Unbiased estimator, but since the test is optional, and even then the simulation only needs

18

to be performed under the null φ = 1, it is not as computationally demanding as the Median Unbiased estimator. It has become common in applied time series econometrics to “err on the side of caution,” by imposing a unit root whenever a unit root cannot be rejected. The blue (upper) line in Figure 7 above shows the simulated RMS standard error for the trend slope when this is done, i.e. when φ = 1 is imposed whenever r1 is greater than 0.7777, its 5% critical value for this regressor matrix and sample size. Although this strategy completely eliminates the small downward bias in the Moment Ratio standard error for φ > .979, it greatly aggravates its moderate upward bias for most other values of φ above approximately 0.6. Such a strategy would therefore be a big mistake. A good case could be made, however, for imposing the unit root whenever exceeds the value of φ for which the bias in the Moment Ratio standard errors changes sign. This value is approximately 0.979 in a trend regression with n = 100, and 0.989 in a trend regression with n = 200.

MRφ

5. Higher Order Autoregressive/Unit Root Models and Consistent Covariance Estimation

Although an AR(1) model is often a good first approximation to the autocovariation function, there is sometimes evidence of higher order serial correlation. This may not truly be a finite order autoregressive process, but a finite order AR(p) model

(29) ∑=

− +=p

jtjtjt u

1εφε

can approximate any stationary Γ to any desired precision, given a high enough value of p. The Yule-Walker equations then determine the autocorrelation function in terms of the φj, and therefore the autocorrelation matrix R.

Define the Moment Ratio Function for rj by pjjpj K,1),(/tr)(tr);,...(ψ 01 == MRMMRMXφφ . The Moment Ratio Function for each rj depends on all the φj through the Yule-Walker equations and therefore R. However, setting pjr pjj KK ,1),;,(ψ 1 == Xφφ , we have a well-conditioned system of p equations in p unknowns that may be solved numerically for as a function of rMR

jφ 1, ... rp, subject to the condition that the

autoregressive roots lie on or outside the unit circle. Then and may be constructed as in (17) and (19).

MR0γ

MRC

The autocovariation function and therefore the coefficient covariance matrix may then be estimated consistently by considering values of the autoregressive order p up to and including a value such as ⎣ ⎦5.075.0 3/1

max −= np ,

19

the maximum lag considered by the “benchmark” formula (22) recommended by Stock and Watson (2007) for HAC.10 Parsimony may be enforced with a general-to-specific model selection procedure that starts with p = pmax and tests the hypothesis φp = 0 at some appropriate test size, say .05, sequentially reducing p by 1 if the hypothesis cannot be rejected. Alternatively, we may simply set p = pmax.

The higher-order Moment Ratio estimators discussed in this section have not yet been implemented. Accordingly, the following two examples are confined to the AR(1) case. 6. Real Income Trend Growth

1950 1960 1970 1980 1990 2000 2010-1.6

-1.4

-1.2

-1

-0.8

-0.6

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0

0.2Real GDP, 1959Q1 - 2008Q4, with trend line

Nat

ural

logs

, all

varia

bles

0 in

last

per

iod

Figure 8 Log US Real GDP, with OLS trend line

Figure 8 above depicts US Real GDP for 1959Q1 – 2008Q4, along with the OLS trend line.11 As indicated in Table 1, the slope of the trend line corresponds to 10 As noted in Footnote 2 above, the formula ⎣ ⎦9/2)100/(4 nm = has very similar effect for n in the range 50 – 2000. 11 Seasonally adjusted quarterly annual rates, chained 2000 dollars, through the “preliminary” values for 2008Q4 released at the end of Feb. 2009. St. Louis Fed FRED data base series GDPC96.

20

3.154%/yr, with an OLS standard error of 0.017%/yr. However, there is acute serial correlation (r1 = 0.946), which makes the OLS standard errors invalid.

Table 1 log yt = a + b (t/4) + εt.

1959Q1 – 2008Q4 (n = 200). DW = 0.059, r1 = 0.946, s2 = 0.00125

MRφ = 0.981, = 0.00200, = 0.0000735 MR0γ

2ˆ uσ

Standard errors coefficient

OLS estimate OLS AR(1) MR

a 0.0767 0.0050 0.0264 0.0448 100b (%/yr) 3.154 0.017 0.089 0.140

The conventional AR(1) standard error, using r1 to estimate φ and s2 to estimate γ0, is 0.089%/yr., which is already 5 times higher than the OLS standard error. However, the Moment Ratio estimate of φ is 0.981. Although this is “not far” from r1 = 0.946, it is less than half as far from unity, and therefore represents more than twice as much persistence. Furthermore, the MR estimate of γ0 is twice as large as s2. Using these two values, the Moment Ratio standard error rises to 0.140%/yr. Although this is 8 times the OLS standard error, the slope remains significantly different from zero (t = 22.5). Although is less than unity, a unit root cannot be rejected using a Monte Carlo simulation of the distribution of r

MRφ1 with Gaussian errors under the null φ = 1. The

actual value, 0.946, is the 49.8 percentile of this simulated distribution using 10,000 replications, so that 0.498 is the p-value for a one-sided unit root test. As noted in Section 4, standard errors based on do not become downward biased in a trend regression with 200 observations until φ exceeds approximately 0.989. Since is only 0.981, there is therefore no compelling reason to impose the non-rejected unit root. In fact, prematurely imposing it can severely bias the standard errors upward. Nevertheless, the two values are close enough that it is worth at least tentatively imposing the unit root.

MRφ

MRφ

Table 1a below imposes the unit root, if only for comparison to Table 1. The unconditional standard error of the intercept becomes infinite, not because there is any great uncertainty about the location of the trend line relative to the data, but simply because there is infinite uncertainty about the location of the data relative to the (undefined) unconditional mean of the error process. However, because the estimated slope depends only on the data and not at all on the unknowable unconditional mean of the error process, its standard error changes only by a finite, though sizeable, amount when the unit root is imposed. Even the intercept has finite standard error when conditioned on the initial or one-step-ahead value of the error.

21

Table 1a log yt = a + b (t/4) + εt.

1959Q1 – 2008Q4 (n = 200). Unit root imposed:

φ = 1, 0γ = ∞, = 0.0000744 2ˆ uσ

Standard errors coefficient

Unconditional | ε0 = 0 | εn+1 = 0

a ∞ 0.1297 0.0449 100b (%/yr) 0.267 0.267 0.267

A unit root in the errors is therefore not equivalent to a spurious relationship between the variables in question. Here we have a highly significant and valid relationship between real GDP and the time trend, that allows us to predict GDP well into the future, even when we assume there is no cointegrating relationship between the two variables. With a unit root, the intercept of the trend line has infinite unconditional uncertainty, but this is completely irrelevant for forecasting GDP, since in the unit root case, there is no tendency for the process to return to the trend line anyway. Instead, it simply follows a random walk from its well observed terminal value, with a growth rate governed by the slope of the trend line, but without reference to the location of the trend line itself. Since this random walk has finite variance innovations, the forecast has finite variance (conditional on the observed data) at all horizons.12 The actual process for real income is likely more complex than a near-unit-root AR(1), if only because of time-aggregation considerations. The present results are merely illustrative of the Moment Ratio estimator, and are not to be taken as a definitive model of real income. 7. Demand for Base Money The near doubling of the quantity of Monetary Base supplied by the Federal Reserve System during the fourth quarter of 2008 makes it important to understand the demand for Base Money. However, this explosion was accompanied by the payment, for the first time, of interest by the Fed on bank excess reserve deposits. The functional form of the demand for Monetary Base may therefore be somewhat different since that quarter than it was previously. Nevertheless, it remains useful to estimate demand for Monetary Base was determined before this change in policy, if only as a proxy for the demand for currency. It is assumed that observed real Base Money balances log mt equal desired real Base Money balances plus (in logs) an error εt that is normal but may be serially correlated. Desired real money balances are assumed to depend on real income yt and

12 In fact, since the standard deviation of the random walk uncertainty only grows with the square root of horizon, while the standard deviation of the growth rate uncertainty grows in proportion to horizon, the latter must dominate at long horizons.

22

nominal interest rates Rt, with a constant real income elasticity a, and a constant interest semielasticity b: log mt = c + a log yt + b Rt + εt. (30) Inventory models of money demand predict that a lies in the range (.5, 1), and that b is negative. The regressors are assumed for the purposes of the present study to be exogenous and thus independent of the errors, though it may be appropriate to revisit this assumption in future work. For this equation, the nominal base is the seasonally adjusted Board of Governors Monetary Base, adjusted for changes in reserve requirements.13 This is deflated by the seasonally adjusted chain-type GDP Deflator.14 Real income is the same real GDP series used in the previous section. The nominal interest rate is the 3-month Treasury bill rate, in %/yr.15 Since real income has a strong trend, at least its coefficient can be estimated consistently by OLS. Since the payment of interest on reserves may have changed the nature of base demand in 2008Q4, the regression is only run for 1959Q1 through 2008Q3. In order for the intercept to indicate the excess demand for Real Base balances at the end of the regression period, all variables are taken relative to their terminal (2008Q3) values. Figure 9 below shows actual log Real Base balances, along with the fitted log demand for Real Base balances. Table 2 below gives the regression results.

13 St Louis Fed FRED data base monthly series BOGAMGSL aggregated to quarterly averages. 14 FRED series GDPCTPI. 15 Secondary market, FRED monthly series TB3MS aggregated to quarterly averages.

23

1950 1960 1970 1980 1990 2000 2010-1.6

-1.4

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0.2Real Monetary Base Demand

Nat

ural

Log

s, a

ll va

riabl

es z

ero

in la

st p

erio

d

ActualFitted

Figure 9

Log Real Monetary Base and estimated log demand for real monetary base. All variables normalized to 0 in last quarter.

Table 2 log mt = c + a log yt + b Rt + εt

mt = real Monetary Base yt = real GDP

Rt = 3-month Treasury bill interest rate All variables normalized to 0 in final quarter

1959Q1 – 2008Q3 (n = 199) DW = 0.142, r1 = 0.921, s2 = 0.00608

MRφ = 0.971, = 0.01317, = 0.000759 MR0γ

2ˆ uσ

Standard errors coefficient

OLS estimate OLS AR(1) MR

c 0.0110 0.0128 0.0543 0.1065 a 0.8618 0.0122 0.0539 0.1070 b -0.0387 0.0020 0.0074 0.0122

24

Again, there is strong serial correlation ( = 0.971) that makes the Moment Ratio standard errors much larger than the downward-biased OLS and even AR(1) standard errors. However, the factor by which the standard error increases is not as strong for the coefficient on R

MRφ

t (6.1) as it is for log(yt) (8.8). The reason for this is that the interest rate is not as strongly serially correlated as is real income, and hence there is not as much interaction with the error serial correlation, as understood already by Bartlett (1935) and Quenouille (1952).

As expected, the income elasticity is positive, and lies in the range (.5, 1). A zero coefficient may easily be rejected, even after the MR correction for serial correlation (t = 8.05), though a unit elasticity may not be rejected (t = 1.29). Also as expected, the interest semi-elasticity is negative. Although not as strong as the income elasticity, it is significantly non-zero (t = -3.17).

As in the previous example, is so high that a unit root cannot be rejected. In a Monte Carlo simulation with 10,000 replications, the actual value of r

MRφ1 was the 42.4

percentile of the simulated distribution of r1, using the X matrix of this regression with φ = 1. The p-value of a one-sided unit root test is therefore 0.424.

Although this is not a pure trend-line regression, log(yt) as illustrated in Figure 8

does not differ greatly from a linear trend, and hence the properties of the regression are not much different from a trend regression. As noted in Section 4 above, the bias in MR trend slope standard errors is actually positive unless φ is greater than approximately 0.989 for n = 200. In the present example, one of the regressors is essentially a linear trend, and the sample size is virtually 200, so a very similar threshold would apply here. Since is only 0.971, imposing a unit root is therefore more likely to bias the standard errors upwards than to correct a downward bias.

MRφ

Nevertheless, the two values are again quite close, so Table 2a below imposes a unit root, if only for comparison to Table 2. Despite the increase in the standard errors, the t-statistics on the income elasticity and interest semi-elasticity are still quite strong (3.36 and -2.30, resp.). There is thus no reason to regard this OLS regression as spurious, even when the errors are assumed to contain a unit root.

25

Table 2a

log mt = c + a log yt + b Rt + εt mt = real Monetary Base

yt = real GDP Rt = 3-month Treasury bill interest rate

All variables normalized to 0 in final quarter 1959Q1 – 2008Q3 (n = 199)

Unit root imposed: φ = 1, γ0 = ∞, = 0.000695 2ˆ uσ

Standard errors

coefficient

Unconditional | ε0 = 0 | εn+1 = 0 c ∞ 0.3800 0.1162 a 0.2563 0.2563 0.2563 b 0.0169 0.0169 0.0169

If the constant semi-elasticity functional form may be extrapolated to out-of-sample interest rates (admittedly a big “if”), the negative reciprocal of the interest rate coefficient, -1/b = Rmax = 25.8%/yr, is the interest rate, and therefore roughly the inflation rate and base expansion rate in excess of real growth, at which real seigniorage is maximized (Cagan 1956). Any seigniorage target above this rate is not consistent with any finite inflation rate, and hence leads eventually to runaway hyperinflation as agents revise their inflationary expectations upward without bound (McCulloch 1982).

If base demand and supply and demand were in equilibrium in 2008Q3, and if interest rates were near zero (the actual 1.49% is close enough for this purpose, given the small values of b and R2008Q3), the maximum theoretically sustainable seigniorage is approximately smax = Rmaxexp(-1) = 9.5% of 2008Q3 monetary base, or about (.095)($861 billion) = $82 billion (2008 dollars). The actual increase of the base was approximately 10 times this value, just in the fourth quarter of 2008.

Since all variables had their terminal values subtracted out before the regression was run, the intercept c indicates that the demand for real base at the 2008Q3 values of y and R exceeded the actual real base in that quarter by 1.10%. This would indicate that in fact steady- state maximal seigniorage is about 1.1% higher than calculated in the preceding paragraph, or about $83 billion (2008 dollars).16 The estimates of base demand in this section are merely designed to illustrate the Moment Ratio approach to estimating autoregressive coefficients. They tell us nothing 16 Although b is significantly non-zero, Rmax and smax could differ substantially from their point estimates. The present paper does not attempt to quantify this uncertainty.

26

about the important dynamics by which the price level adjusts to clear any excess supply or demand for base money, nor by which base demand adjusts to real income and/or interest rates. Furthermore, there may be higher order serial correlation present in addition to the first order modeled here. 17

8. Conclusions The proposed Moment Ratio Estimator for the autoregressive parameters of the errors in an OLS regression is computed from the conventional residual autocorrelation coefficients, but greatly reduces their bias, and provides corrected standard errors with far less bias than alternatives. The estimator is in the spirit of the Median Unbiased estimator of Andrews (1993) and McCulloch (2008), but does not require their Monte Carlo simulation, and provides smaller standard error bias in the illustrated case. The presence of a unit root in the errors, and therefore the absence of a cointegrating relationship, does require reposing the problem, but does not by itself indicate that an OLS correlation between the variables is spurious. Hypothesis testing is standard, provided it is based on squared quasi-differenced residuals, and not on the squared residuals themselves, and provided the regressor(s) include a time trend or unit root regressor with non-zero drift. Although the present paper is restricted to the AR(1) case, the approach is readily extendable to higher-order AR processes. An exact unit root test similar to that of Andrews (1993) is implemented for the AR(1) case. The Moment Ratio estimator is applied to an income trend line regression, as well as to a monetary base demand function. In both cases, the Moment Ratio autoregressive coefficient estimate is quite close to unity, and a unit root in the errors cannot be rejected. However, the trend slope remains highly significant in the income trend line regression, and both the income elasticity and interest semi-elasticity remain highly significant in the base demand equation, even when a unit root is imposed.

In general, a unit root should not be imposed whenever it cannot be rejected, since unless the autoregressive coefficient is very close to unity, falsely imposing a unit root will bias coefficient standard errors upwards. Despite their consistency, the popular HAC standard errors of Newey and West (1987) can greatly overstate the precision of OLS coefficient estimates with sample sizes and serial correlation commonly found in economic studies when, as has become standard, “automatic bandwidth selection” is employed.

17 Adding leads and lags of the first differences of the regressors, as in Choi, Hu and Ogaki (2008), may be a useful way to enrich the dynamics of the regression.

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REFERENCES

Andrews, Donald W.K.. 1993. Exactly Median-Unbiased Estimation of First Order Autoregressive / Unit Root Models. Econometrica 61:139-65. Andrews, Donald W.K., and J. Christopher Monahan. 1992. An Improved Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimator. Econometrica 60:953-66. Bartlett, M.S. 1935. Some aspects of the time-correlation problem in regard to tests of significance. Journal of the Royal Statistical Society 98: 536-43. Cagan, Philip. 1956. The monetary dynamics of hyperiinflation. in M. Friedman, ed., Studies in the Quantity Theory of Money. Chicago: Univ. of Chicago Press. Choi, Chi-Young, Ling Hu, and Masao Ogaki. 2008. Robust Estimation for Structural Spurious Regressions and a Hausman-type Cointegration Test. Journal of Econometrics 142: 327-351. Den Haan, Wouter J., and Andrew Levin. 1997. A Practitioner’s Guide to Robust Covariance Matrix Estimation. In G.S. Maddala and C.R. Rao, eds., Robust Inference (Handbook of Statistics 15). Elsevier. Greene, William H. 2003. Econometric Analysis, 5th ed. Prentice Hall. Hayashi, Fumio. 2000. Econometrics. Princeton Univ. Press. McCulloch, J. Huston. 1982. Money and Inflation: A Monetarist Approach, 2nd ed. New York: Academic Press. __________. 2008. Median-Unbiased Estimation of Higher Order Autoregressive/Unit Root Processes and Autocorrelation Consistent Covariance Estimation in a Money Demand Model. Presented at Econometric Society 2008 North American Summer Meetings, Pittsburgh, June 19-22, and at 14th International conference on Computing in Economics and finance, Paris, June 26-28, Online at <http://www.econ.ohio-state.edu/jhm/papers/MUARM1S.pdf> Newey, Whitney, and Kenneth West. 1987. A Simple Positive Semi-Definite Heteroskedastic and Autocorrelation Consistent Covariance Matrix.” Econometrica 55:703-8. Quenouille, M.H. 1952. Associated Measurements. London: Butterworths Scientific Publications. Stock, James H., and Mark W. Watson. 2007. Introduction to Econometrics, 2nd ed. Pearson/Addison-Wesley.

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Appendix

The following page shows first, the correlation matrix R for n = 100 and φ = .98, and second, the matrix W0 that replaces it in the unit root case φ = 1.

The next page shows first, the covariance matrix ZRZ of demeaned errors with φ = .98 and unit error variance, where Z is the mean-removing matrix defined in (28). Notice that the residuals all have variance less than (n-k)/n, and are heteroskedastic. Because the variance of the residuals is highest for the two ends of the sequence, and the covariance is negative from one end of the sequence to the other, there will be a tendency for a spurious up- or down-trend to appear to be present in the demeaned data. This may easily appear to be significant using OLS standard errors, but MR-corrected standard errors will give correct size.

The second image on that page is the covariance matrix ZW0Z of de-meaned errors with φ = 1, and unit innovation variance. Apart from the different convention regarding vertical scale, the two images are almost indistinguishable. Although it is evident from W0 which error it conditions on, the properties of the demeaned errors do not depend on this choice. The third page shows the covariance matrix for detrended residuals with φ = .98 and φ = 1, i.e. MRM and MW0M, where M is computed from the trend line regressor matrix X = [ones(n,1) (1:n)’] (in MATLAB notation). Again, the two matrices are almost indistinguishable apart from vertical scale convention, and the second covariance matrix does not depend on the choice of Wt. The last page shows the nominal monetary base through 2008Q4. The last point was excluded from the regression in Section 7 above.

29

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