Demand for Net Monetary Base

With Moment Ratio-based

Standard Errors

J. Huston McCulloch

Ohio State University

US Monetary Base and Net Base = Base – interest-bearing XS Reserves

If XS reserves pay market interest rate, they represent financial intermediation and have no inflationary wealth effect.

log mt = c + a log yt + b Rt + t.

mt = Real Net Base yt = Real GDPRt = 3-mo T-Bill rate

Data via St. Louis Fed FRED data base

Nominal net base deflated with GDP deflator

All variables normalized to 0 in last quarter so that c measures excess D for m in last quarter (2011Q1).

A simple net base demand function

Data: Real GDP

Data: 3-Mo T-Bill Rate

Near 0 since 2009.Markets distorted 1980Q1 by Carter Credit Controls – T-bill rates shot down to 8%, Prime rate up to 20%+.

OLS regression results 1959Q1 – 2011Q1 (n = 209)

coef. OLS est. Standard errors(t-statistics)

OLS

c -0.0632 0.0120(-5.26)

a 0.8598 0.0111(77.30)

b -0.0391 0.0018(-21.37)

Standard errors small, t-stats huge!

OLS regression results 1959Q1 – 2011Q1 (n = 209)

coef. OLS est. Standard errors(t-statistics)

OLS

c -0.0632 0.0120(-5.26)

a 0.8598 0.0111(77.30)

b -0.0391 0.0018(-21.37)

Standard errors small, t-stats huge! But – DW = 0.154, p = 9.4e-92!

So OLS standard errors invalid

OLS regression results 1959Q1 – 2011Q1 (n = 209)

coef. OLS est. Standard errors(t-statistics)

OLS

c -0.0632 0.0120(-5.26)

a 0.8598 0.0111(77.30)

b -0.0391 0.0018(-21.37)

Standard errors small, t-stats huge!

Low-Tech solution: ignore problem

OLS regression results 1959Q1 – 2011Q1 (n = 209)

coef. OLS est. Standard errors(t-statistics)

OLS HAC(5)

c -0.0632 0.0120(-5.26)

0.0209(-3.03)

a 0.8598 0.0111(77.30)

0.0215(39.93)

b -0.0391 0.0018(-21.37)

0.0040(-9.78)

HAC se’s bigger, but t-stats still plenty big! Now “standard” correction for serial correlation.Easy radio button in EViews etc.

High-Tech solution: Use Newey-West HAC standard errors

Actual vs Predicted real Net Base

Long runs of +, - errors indicate positive serial correlation

Residuals persistent but appear to be stationary

But regression residuals typically less persistent, have smaller variance than true errors.

εXβy

||)'E( ji εεΓ

εXXXβyXXXβ ')'(')'(ˆ 11

11

11

)'(')'(

)'('')'(E

)ˆ(

XXΓXXXX

XXXεεXXX

βCovC

0|| / ΓR ji

OLS Regression with non-spherical stationary errors

X exogenous, includes const.

C depends on all autocovariances j

12 )'(ˆ XXC sOLS IΓ 0unbiased only if

MεεXXXXIβXye )')'((ˆ 1

1,,0)(tr1

njees j

jn

ijiij ee'

)),((

)(tr1

,

jdiagsum

ajn

ijiij

A

A

1,,0,/ 0 njssr jj

,)'E(E MΓMMεεMee'

)(tr)(trE 0 MRMMΓM jjjs

Residuals and Sample Autocorrelations

j-th order trace:

Newey-West HAC standard errors HAC = Heteroskedasticity and Autocorrelation Consistent Now routinely used as “correction” for serial corr.

11 )'(')'(ˆ XXFXXXXCHAC

choicebandwidth automatic 1)100/(4

function kernelBartlett )0,/)max(()k(

|)(|

9/2

nm

mlml

jikee jiF

,

Consistent because m, n/m as n .

But biased downwards with n < for 3 reasons:• Uses only first m-1 autocovariances• Downweights those by Bartlett factor• Uses e’s as if they were ’s -- MM in place of

,1 ttt u Eg AR(1) process = .9

• Higher order autocovariances just noise, so ignore• Lower order autocovariances reflect AR(1) process

• but start off too small,• decay too fast.

• NW is a step in wrong direction (m = 5 illustrated)

AR(p) Standard Errors AR(1) may be too restrictive. Instead, assume errors AR(p):

2

1

~, iiduu t

p

jtjtjt

Yule-Walker eqn’s determine R, G = /2 as a fn. of 1, ... p

and vice-versa.Standard Method of Moments estimates i by ri.

1 - mbandwidth NW )100/(4Set 9/2 np

so as to use same lags as NW without truncation or down-weighting.

OLS regression results 1959Q1 – 2011Q1 (n = 209)

coef. OLS est. Standard errors(t-statistics)

OLS HAC(5) AR(4)

c -0.0632 0.0120(-5.26)

0.0209(-3.03)

0.0375(-1.69)

a 0.8598 0.0111(77.30)

0.0215(39.93)

0.0361(23.82)

b -0.0391 0.0018(-21.37)

0.0040(-9.78)

0.0051(-7.40)

AR(4) SEs bigger, t-stats smaller But AR(4) SE’s still downward biased

jjttt u ,1

MM vs. in AR(1) model

0 20 40 60 80 100

0

50

1000.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Cov(), = 0.98, n = 100, var(t) = 1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 20 40 60 80 100

0

50

100-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Cov(demeaned residuals), = 0.98, n = 100, var(t) = 1

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0 20 40 60 80 100

0

50

100-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Cov(detrended residuals), = 0.98, n = 100, var(t) = 1

-0.1

0

0.1

0.2

0.3

0.4

True

MM (constant only) MM (trendline)

Residuals much less persistentthan errors themselves.

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

1

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

Mon

te C

arlo

dis

tribu

tion

of r

1 = s

1/s0,

(,X

)=E

s1/E

s0

95%75%(,X)median

mean

25%5%

Monte Carlo Distribution of r1 in AR(1) model

Bias becomes acute as approaches 1!Bias similar for total persistence in AR(p) model

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

1

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

true

(

,X)

= E

s 1/Es 0

(,X)

45 deg

Moment Ratio Estimator in AR(1) case r1 = s1/s0 is ratio of 2 sample moments

Moment Ratio Function:

is ratio of population moments consistently est. by s1, s0.

)(/tr)(tr);(ψ 011 MRMMRMX

Moment Ratio Estimator -- AR(1) case

)(/tr)(tr);ψ( 011 MRMMRMXr );(ψˆ1

1 XrMR

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 (trend line model, n = 100)

r1 = s

1/s

0

hat MR

-1(r1,X)

45 deg

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

1

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

Mon

te C

arlo

dis

tribu

tion

of r

1 = s

1/s0,

(,X

)=E

s1/E

s0

95%75%(,X)median

mean

25%5%

Moment Ratio function Monte Carlo median

MR Estimator approximately median unbiased without costly simulation of Andrews (1993).

pp

pMM

s

s

ss

ss

1

1

01

10

φ

.

tr

tr

trtr

trtr

E

E

EE

EE

;1

1

01

101

1

01

10

MGM

MGM

MGMMGM

MGMMGM

Xφψ

pp

p

pp

p

s

s

ss

ss

Xφψφφφ

,ˆminargˆ

MMMR

MR(p) Estimator

so define

then numerically solve

Constrained Nelder-Mead sol’n of MR eq’ns:

N = 100, p = 4, tol = .001: 107 iterations, 0.4 sec on ordinary laptop.Circles = AR(p) starting point, boxes = MR(p) sol’n.

Unit Root does not imply spurious regression! but requires reformulating problem. ADF / Andrews & Chen (94) persistence form w/ = 1:

p

jhhj

p

jjt

p

jjtjtjtt u

11

1

111 .,

2211 /,Cov ttttHYule-Walker gives

matrix integrator is ,)|Cov( 20 ji NNHN'ε

MNHNMee 20 Cov|Cov

pjs jj ,1),(trE 2 MNHNM

Monte Carlo bias, size distortion of MR(p) trendline regression, n = 100, p = 4, AR(1) DGP, 10,000 reps

Median squared SE of slope coefficient / true variance:

MR(p) dominates AR(p), HAC, OLS i.t.o. median biasUnless very near 0

Coverage of 95% CI for slope (full graph)

Coverage of 95% CI for slope (detail of previous slide)

MR(p) outperforms others.Use Student t with reduced DOF?

MMj

MRjlag j ri

1 0.913 1.117 1.159

2 0.808 -0.393 -0.433

3 0.735 0.405 0.453

4 0.639 -0.235 -0.251

0.895 0.927

MR vs MM AR(4) coefficient estimates:

MR raises persistence, but still short of unit root.

OLS regression results 1959Q1 – 2011Q1 (n = 209)

coef. OLS est. Standard errors(t-statistics)

OLS HAC(5) AR(4) MR(4)

c -0.0632 0.0120(-5.26)

0.0209(-3.03)

0.0375(-1.69)

0.0517(-1.22)

a 0.8598 0.0111(77.30)

0.0215(39.93)

0.0361(23.82)

0.0503(17.08)

b -0.0391 0.0018(-21.37)

0.0040(-9.78)

0.0051(-7.40)

0.0071(-5.54)

MR(4) se’s bigger than AR(4), but a, b still significant! However, c, although large, is insignificant.

Issues for future work:

• Implement Unit Root test,• Find rule for when to impose unit root

• Properties with Long Memory errors?

• Regressor-Conditional Heteroskedasticity • White / NW-type modification?

0 20 40 60 80 100

0

50

100-10

-5

0

5

10

15

Cov(detrended residuals), = 1, n = 100, var(ut) = 1

-4

-2

0

2

4

6

8

10

12

Thank you! Questions?

The AR(1) Unit Root case = 1

02

00 )|'E( WεεΓ u

),max(0 jiW

MWMeeee 02

0 )|'E('E u

)(trE 02 MWMjujs

)(/tr)(tr);1ψ( 0001 MWMMWMX

)(tr/ˆ 0002 MWMsu

0 20 40 60 80 100

0

50

1000

20

40

60

80

100

Cov(|0 = 0), = 1, n = 100, var(u

t) = 1

10

20

30

40

50

60

70

80

90

100

0 20 40 60 80 100

0

50

100-20

-10

0

10

20

30

40

Cov(demeaned residuals), = 1, n = 100, var(ut) = 1

-15

-10

-5

0

5

10

15

20

25

30

0 20 40 60 80 100

0

50

100-10

-5

0

5

10

15

Cov(detrended residuals), = 1, n = 100, var(ut) = 1

-4

-2

0

2

4

6

8

10

12