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

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. A simple net base demand function log mt = c + a log yt+ b Rt + t. mt = Real Net Base yt = Real GDP Rt = 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).

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) Standard errors small, t-stats huge! 

7. OLS regression results 1959Q1 – 2011Q1 (n = 209) 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) Low-Tech solution: ignore problem Standard errors small, t-stats huge! 

9. OLS regression results 1959Q1 – 2011Q1 (n = 209) High-Tech solution: Use Newey-West HAC standard errors HAC se’s bigger, but t-stats still plenty big!  Now “standard” correction for serial correlation. Easy radio button in EViews etc.

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.  = .9 Eg AR(1) process • 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 ri. so as to use same lags as NW without truncation or down-weighting.

17. OLS regression results 1959Q1 – 2011Q1 (n = 209) AR(4) SEs bigger, t-stats smaller But AR(4) SE’s still downward biased 

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

19. Monte Carlo Distribution of r1 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 • r1 = s1/s0 is ratio of 2 sample moments • Moment Ratio Function: • is ratio of population moments consistently est. by s1, s0.

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. MR vs MM AR(4) coefficient estimates: MR raises persistence, but still short of unit root.

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