1. Demand for Net Monetary Base With Moment Ratio-based Standard Errors J. Huston McCulloch Ohio State University

2. 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.

3. log m t = c + a log y t + b R t +  t. m t = Real Net Base y t = Real GDP R t = 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

4. Data: Real GDP

5. 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%+.

6. OLS regression results 1959Q1 – 2011Q1 (n = 209) coef.OLS est.Standard errors (t-statistics) OLS c-0.06320.0120 (-5.26) a 0.85980.0111 (77.30) b-0.03910.0018 (-21.37) Standard errors small, t-stats huge!

7. OLS regression results 1959Q1 – 2011Q1 (n = 209) coef.OLS est.Standard errors (t-statistics) OLS c-0.06320.0120 (-5.26) a 0.85980.0111 (77.30) b-0.03910.0018 (-21.37) Standard errors small, t-stats huge! But – DW = 0.154, p = 9.4e-92! So OLS standard errors invalid 

8. OLS regression results 1959Q1 – 2011Q1 (n = 209) coef.OLS est.Standard errors (t-statistics) OLS c-0.06320.0120 (-5.26) a 0.85980.0111 (77.30) b-0.03910.0018 (-21.37) Standard errors small, t-stats huge! Low-Tech solution: ignore problem

9. OLS regression results 1959Q1 – 2011Q1 (n = 209) coef.OLS est.Standard errors (t-statistics) OLSHAC(5) c-0.06320.0120 (-5.26) 0.0209 (-3.03) a 0.85980.0111 (77.30) 0.0215 (39.93) b-0.03910.0018 (-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

10. Actual vs Predicted real Net Base Long runs of +, - errors indicate positive serial correlation

11. Residuals persistent but appear to be stationary But regression residuals typically less persistent, have smaller variance than true errors.

12. OLS Regression with non-spherical stationary errors X exogenous, includes const. C depends on all autocovariances  j unbiased only if

13. Residuals and Sample Autocorrelations j-th order trace:

14. Newey-West HAC standard errors HAC = Heteroskedasticity and Autocorrelation Consistent Now routinely used as “correction” for serial corr., 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 -- M  M in place of 

15. 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)

16. AR(p) Standard Errors AR(1) may be too restrictive. Instead, assume errors AR(p): Yule-Walker eqn’s determine R, G =  /  2 as a fn. of  1,...  p and vice-versa. Standard Method of Moments estimates  i by r i. so as to use same lags as NW without truncation or down-weighting.

17. OLS regression results 1959Q1 – 2011Q1 (n = 209) coef.OLS est.Standard errors (t-statistics) OLSHAC(5)AR(4) c-0.06320.0120 (-5.26) 0.0209 (-3.03) 0.0375 (-1.69) a 0.85980.0111 (77.30) 0.0215 (39.93) 0.0361 (23.82) b-0.03910.0018 (-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 

18. M  M vs.  in AR(1) model True  M  M (constant only) M  M (trendline) Residuals much less persistent than errors themselves.

19. Monte Carlo Distribution of r 1 in AR(1) model Bias becomes acute as  approaches 1! Bias similar for total persistence in AR(p) model

20. Moment Ratio Estimator in AR(1) case r 1 = s 1 /s 0 is ratio of 2 sample moments Moment Ratio Function: is ratio of population moments consistently est. by s 1, s 0.

21. Moment Ratio Estimator -- AR(1) case

22. Moment Ratio function  Monte Carlo median  MR Estimator approximately median unbiased without costly simulation of Andrews (1993).

23. MR(p) Estimator so define then numerically solve

24. 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.

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

26. 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 bias Unless  very near 0

27. Coverage of 95% CI for slope (full graph)

28. Coverage of 95% CI for slope (detail of previous slide) MR(p) outperforms others. Use Student t with reduced DOF?

29. lag jriri 10.913 1.117 1.159 20.808-0.393-0.433 30.735 0.405 0.453 40.639-0.235-0.251  0.895 0.927 MR vs MM AR(4) coefficient estimates: MR raises persistence, but still short of unit root.

30. OLS regression results 1959Q1 – 2011Q1 (n = 209) coef.OLS est.Standard errors (t-statistics) OLSHAC(5)AR(4)MR(4) c-0.06320.0120 (-5.26) 0.0209 (-3.03) 0.0375 (-1.69) 0.0517 (-1.22) a 0.85980.0111 (77.30) 0.0215 (39.93) 0.0361 (23.82) 0.0503 (17.08) b-0.03910.0018 (-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. 

31. 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?

32. Thank you! Questions?

33. The AR(1) Unit Root case  = 1