USING OUT-OF-SAMPLE MEAN SQUARED PREDICTION ERRORS TO TEST THE MARTINGALE DIFFERENCE HYPOTHESIS. Todd E. Clark Federal Reserve Bank of Kansas City
|
|
- Dorothy Todd
- 7 years ago
- Views:
Transcription
1 USING OUT-OF-SAMPLE MEAN SQUARED PREDICTION ERRORS TO TEST THE MARTINGALE DIFFERENCE HYPOTHESIS Todd E. Clark Federal Reserve Bank of Kansas City Kenneth D. West University of Wisconsin
2 Example: predicting monthly changes in Swiss franc/dollar exchange rate Null model: random walk (predicts value of 0 for next month s change in the exchange rate) Alternative model: use interest differential to predict y t+1 = % change in franc/dollar rate in month t+1 i t -i * t = U.S. interest rate - Swiss interest rate, monthly Predict using OLS estimates of y t+1 = b 0 +b(i t -i * t ) + e t+1
3 To estimate y t+1 = b 0 +b(i t -i * t ) + e t+1, we use 10 year rolling regressions ( R = 120 months), with the first sample beginning in 1975: estimate 1975:1-1984:12, predict 1985:1 estimate 1975:2-1985:1, predict 1985:2... estimate 1993: :9, predict 2003:10 So predictions run from 1985:1-2003:10 (number of predictions P = 226) Figure (plot of ^b t, t=1984: :9)
4 Mean squared prediction error (MSPE) from random walk model : 9 t = 1985:1 y2 t+1 /226 = MSPE from interest parity model = G 2 t 003: 9 = 198 5:1 [y t+1 - ^b 0t - ^b t (i t -i * t )]2 /226 = As in many earlier studies, random walk model has smaller MSPE (i.e., 12.27<12.33). We argue: one can nonetheless reject at the 5% level the null the random walk model predicts as well as the interest parity model by a MSPE criterion, against the alternative that the interest parity model predicts better.
5 Intuition: If the random walk model is right, the alternative model (the interest parity model) introduces noise into the forecasting process: the alternative model attempts to estimate parameters b 0 and b that are zero in population. In finite samples, use of the noisy estimate of the parameter will raise the estimated MSPE of the interest parity model relative to the random walk model. So if the random walk model is right, the random walk MSPE should not only be smaller but be smaller by the amount of estimation noise.
6 That noise is estimable (details below), and one can and should adjust MSPE comparisons for the estimated noise. In this example, the estimate happens to be So ^F 1 2 = MSPE for random walk = ^ F2 2 = MSPE for interest parity = adj. = adjustment for estimation noise = 0.96 ^ F2 2 -adj. = MSPE for interest parity, adjusted for estimation noise = = Point estimate: 11.37<12.27, i.e., adjusted MSPE for interest parity model is less than that of random walk model One can construct in familiar fashion a standard error for a test of H 0 : After adjustment for estimation noise, MSPE for interest parity = MSPE for random walk (Diebold and Mariano (1995), West (1996))
7 Previous literature on MSPE comparisons for nested models: West (1996) notes specifically that his procedures maintain a rank condition that is not satisfied when models are nested; a similar rank condition is implicit in Diebold and Mariano (1995) McCracken (2000) and Clark and McCracken (2001, 2003) present analytical and simulation evidence to argue vigorously that standard critical values should not be used when models are nested Some applied papers therefore use simulations to get critical values (e.g., Mark (1995), Kilian and Taylor (2003)) Other applied papers use standard critical values, sometimes with apology (Clarida et al. (2003), Cheung et al. (2003)) by construction, would not reject null in Swiss franc example, since MSPE for random walk < (unadjusted) MSPE for interest parity
8 Some theoretical papers have proposed alternative tests for nested models Chao, Corradi and Swanson (2003) propose an encompassing test Giacomini and White (2003) propose a certain conditional test Our procedure - adjust the alternative model s MSPE for bias from estimation, then compare to the null model using familiar procedures - implicitly transforms the MSPE comparison to an encompassing test, though not the test proposed by Chao et al. (2001)
9 I. Introduction and background II. This paper s procedure III. Simulation evidence IV. Empirical example V Conclusions
10 II. This paper s procedure Specifically, (2.1) y t = e t (model 1: null model), (2.2) y t = X t N$+e t (model 2: alternative model). (2.4) (2.5) ^ F 2 1 / P-1 G T t=t-p+1 y2 t+1 = MSPE from model 1, ^ F 2 2 / P-1 G T t=t-p+1 (y t+1 -X t+1 N ^$ t ) 2 = MSPE from model 2. Since we have y 2 t+1 - (y t+1 -X t+1 N ^$ t ) 2 = 2y t+1 X t+1 N ^$ t - (X t+1 N ^$ t ) 2 (2.6) ^ F ^F 2 2 = 2(P-1 G T t=t-p+1 y t+1 X t+1 N ^$ t ) - [P -1 G T t=t-p+1 (X t+1 N ^$ t ) 2 ].
11 (2.6) ^ F ^F 2 2 = 2(P-1 G T t=t-p+1 y t+1 X t+1 N ^$ t ) - [P -1 G T t=t-p+1 (X t+1 N ^$ t ) 2 ]. ==== 2(P -1 G T t=t-p+1 y t+1 X t+1 N ^$ t ). 0, - [P -1 G T t=t-p+1 (X t+1 N ^$ t ) 2 ] < 0 so under the MDS null we expect ^ F ^F 2 2 <0 or: we expect the MSPE from the MDS model to be less than that from the alternative model. We propose looking not at ^F ^F 2 2 but at ^F ^F 2 2 -adj., where ^ F 2 2 -adj. = ^F [P-1 G T t=t-p+1 (X t+1 N ^$ t ) 2 ].
12 ^ F 2 2 -adj. = ^F [P-1 G T t=t-p+1 (X t+1 N ^$ t ) 2 ] === In the Swiss franc data, ^ F 2 2 = [P -1 G T t=t-p+1 (X t+1 N ^$ t ) 2 ] = 0.96 ^ F 2 2 -adj. = < = ^F 2 1 = MSPE from random walk model
13 Figures 1 and 2
14 Inference (3.3) ^ f t+1 / y 2 t+1 - [(y t+1 -X t+1 N ^$ t ) 2 - (X t+1 N ^$ t ) 2 ]. Then one can compute (3.4) b f = P -1 G T t=t-p+1 Then ^ f t = ^F ^F 2 2 -adj. = ^F ^F [P-1 G T t=t-p+1 (X t+1 N ^$ t ) 2 ]. qp b f - A N(0,V), V consistently estimated by ^V = P -1 G T t=t-p+1 Perform t-test with qp b f / q ^V ^ f t 2.
15 Asymptotics rolling samples used to generate regressions (rolling sample size R=120 in the exchange rate example) P = number of predictions (P=168 in the exchange rate example) Total sample size T+1=R+P As T64, P64 but R stays fixed (Giacomini and White (2003)) versus West and McCracken (1998), McCracken (2000) and Clark and McCracken (2001, 2003): T64, P64, R64. Various mixing and moment conditions (Giacomini and White (2003)); conditional heteroskedasticity allowed; stationarity; parametric linear alternative extensions to allow moment drift and parametric nonlinear alternative are straightforward
16 III. Simulation evidence We examine MSPE-adjusted (our statistic) MSPE: normal MSPE: McCracken (same statistic as MSPE: normal, but different critical values) CCS: a certain encompassing test of Chao, Corradi and Swanson (2001)
17 Two DGPs Both assume alternative model is univariate (X includes a constant and a single stochastic variable x) x is highly persistent (AR(1) with parameter 0.95) y (=presumed martingale difference sequence) much more variable than x
18 DGP 1: intended to capture some central characteristics of monthly exchange rate data little contemporaneous correlation between innovation in x (= interest differential) and innovation in y (=exchange rate) when simulating under alternative, y t+1 = -2x t + shock DGP 2: intended to capture some central characteristics of monthly stock price data high negative correlation between innovation in x (=dividend/price ratio) and innovation in y (=stock return) conditional heteroskedasticity of y in some parameterizations when simulating under alternative, y t+1 =.365x t + shock
19 Summary of simulation results for nominal one-tailed10% tests, one step ahead forecasts MSPE-adjusted has size of 5-11%; across DGPs and various sample sizes, median size is 8% MSPE-normal has size of 0-7%; median size is 0% MSPE-McCracken has size of 2-21%; median size is 9% Chao et al. has size of 10-25%; median size is 13% Size-adjusted power comparable across tests
20 Tables 1-6, Figure 3
21 Unadjusted power (DGP 1, b=-2) (4.6) R=120, P=144 R=120, P=240 MPSE-adjusted MSPE: normal MSPE: McCracken CCS
22 IV. Empirical example Monthly bilateral exchange rates and interest differentials, U.S. vs: Canada, Japan, Switzerland, U.K. Rolling samples of size R=120 months First regression sample is 1975:1-1984:12 for Switzerland and U.K., 1980:1-1989:12 for Canada and Japan Last prediction for all four countries is 2003:12; number of monthly predictions P=166 for Canada and Japan, P = 226 for Switzerland and the U.K.
23 (Unadjusted) MSPE interest differential model is more than that for random walk in Japan, Switzerland, U.K.; interest differential model does beat random walk for Canada Adjusted MSPE for interest differential model is less than that for random walk in all four countries MSPE-adjusted test statistic rejects null in Switzerland and Canada at the 5% level.
24 MSPE- MSPE: adjusted normal ^ F1 2 ^ F2 2 ^ adj. F2 2 -adj ^F2 1 -( ^F 2 2 -adj.) ^F2 1 - ^F 2 2 Canada CCS (0.08) 1.78 ** Japan (0.43) * Swiss (0.48) 1.88 ** U.K (0.33)
25 V. Conclusions When testing the MDS hypothesis, MSPE comparisons should be adjusted for noise from parameter estimation Our procedure is straightforward and works reasonably well
26 Important extensions null model that uses estimated parameter vector to predict recursive (instead of rolling) regressions to generate predictions
27 1 Rolling Coefficient Estimates, Switzerland
28 5.0 Rolling Coefficient Estimates, Switzerland (with 90% confidence band)
29 Figure 1 Density of Simulation MSPEs Under the Null, R=120, P Varying, DGP 1 A. MSPE(1) - MSPE(2) P = 48 P = 144 P = 240 P = B. Average of (X*betahat)^2 P = 48 P = 144 P = 240 P = C. MSPE(1) - [MSPE(2) - average of (X*betahat)^2] P = 48 P = 144 P = 240 P =
30 Figure 2 Density of Simulation MSPEs Under the Null, R Varying, P=144, DGP 1 R = 60 R = 120 R = 240 A. MSPE(1) - MSPE(2) B. Average of (X*betahat)^2 R = 60 R = 120 R = C. MSPE(1) - [MSPE(2) - average of (X*betahat)^2] R = 60 R = 120 R =
31 Figure 3 Density of Simulation MSPE-Adjusted Test Statistic Under the Null, DGP A. R Varying, P= R = 60 R = 120 R = B. R=120, P Varying P = 48 P = 144 P = 240 P = 1200
32 Figure 4 Density of Simulation MSPE Test Statistic (Unadjusted) Under the Null, DGP A. R Varying, P= R = 60 R = 120 R = B. R=120, P Varying P = 48 P = 144 P = 240 P = 1200
33 Figure 1 Density of Simulation MSPEs Under the Null, R=120, P Varying, DGP 1 A. MSPE(1) - MSPE(2) P = 48 P = 144 P = 240 P = B. Average of (X*betahat)^2 P = 48 P = 144 P = 240 P = C. MSPE(1) - [MSPE(2) - average of (X*betahat)^2] P = 48 P = 144 P = 240 P =
34 Figure 2 Density of Simulation MSPEs Under the Null, R Varying, P=144, DGP 1 R = 60 R = 120 R = 240 A. MSPE(1) - MSPE(2) B. Average of (X*betahat)^2 R = 60 R = 120 R = C. MSPE(1) - [MSPE(2) - average of (X*betahat)^2] R = 60 R = 120 R =
35 Figure 3 Density of Simulation MSPE-Adjusted Test Statistic Under the Null, DGP A. R Varying, P= R = 60 R = 120 R = B. R=120, P Varying P = 48 P = 144 P = 240 P = 1200
36 Figure 4 Density of Simulation MSPE Test Statistic (Unadjusted) Under the Null, DGP A. R Varying, P= R = 60 R = 120 R = B. R=120, P Varying P = 48 P = 144 P = 240 P = 1200
37 Table 1 Empirical Size: DGP 1 Nominal Size = 10% A. R=60 P=48 P=96 P=144 P=240 P=480 P=1200 MSPE-adjusted MSPE-normal MSPE-McCracken CCS B. R=120 P=48 P=96 P=144 P=240 P=480 P=1200 MSPE-adjusted MSPE-normal MSPE-McCracken CCS C. R=240 P=48 P=96 P=144 P=240 P=480 P=1200 MSPE-adjusted MSPE-normal MSPE-McCracken CCS Notes: 1. The predictand y t+1 is i.i.d. N(0,1); the alternative model s predictor x t follows an AR(1) with parameter 0.95; data are conditionally homoskedastic. In each simulation, one step ahead forecasts of y t+1 are formed from the martingale difference null and from rolling estimates of a regression of y t on X t = (1,x t-1 )N. 2. R is the size of the rolling regression sample. P is the number of out-of-sample predictions. 3. Our MSPE adjusted statistic, defined in (3.1) and (4.4), uses standard normal critical values. MSPE normal, defined in (4.5), refers to the usual (unadjusted) t-test for equal MSPE, and also uses standard normal critical values. MSPE-McCracken relies on the MSPE-normal statistic but uses the asymptotic critical values of McCracken (2000). 4. The number of simulations is 10,000. The table reports the fraction of simulations in which each test rejected the null using a one-sided test at the 10% level. For example, the figure of.078 in panel A, P=48, MSPE-adjusted, indicates that in 780 of the 10,000 simulations the MSPE-adjusted statistic was greater than 1.65.
38 Table 2 Empirical Size: DGP 2 Nominal Size = 10% A. R=60 P=48 P=96 P=144 P=240 P=480 P=1200 MSPE-adjusted MSPE-normal MSPE-McCracken CCS B. R=120 P=48 P=96 P=144 P=240 P=480 P=1200 MSPE-adjusted MSPE-normal MSPE-McCracken CCS C. R=240 P=48 P=96 P=144 P=240 P=480 P=1200 MSPE-adjusted MSPE-normal MSPE-McCracken CCS Notes: 1. See the notes to Table 1.
39 Table 3 Empirical Size: DGP 2 with Conditional Heteroskedasticity Nominal Size = 10% A. GARCH P=48 P=96 P=144 P=240 P=480 P=1200 MSPE-adjusted MSPE-normal MSPE-McCracken CCS B. Multiplicative conditional heteroskedasticity P=48 P=96 P=144 P=240 P=480 P=1200 MSPE-adjusted MSPE-normal MSPE-McCracken CCS Notes: 1. See the notes to Table 1. The regression sample size R is In the upper panel of results, the predictand y t+1 is a GARCH process, with the parameterization given in equation (4.4). In the lower panel, the predictand y t+1 has conditional heteroskedasticity of the form given in equation (4.5), in which the conditional variance at t is a function of x 2 t-1.
40 Table 4 Size-Adjusted Power: DGP 1 Empirical Size = 10% A. R=60 P=48 P=96 P=144 P=240 P=480 P=1200 MSPE-adjusted MSPE CCS B. R=120 P=48 P=96 P=144 P=240 P=480 P=1200 MSPE-adjusted MSPE CCS C. R=240 P=48 P=96 P=144 P=240 P=480 P=1200 MSPE-adjusted MSPE CCS Notes: 1. The DGP is defined in equation 4.3, with: b=-2; e t+1 ~ i.i.d. N(0,1); x t ~ AR(1) with parameter 0.95; data are conditionally homoskedastic. In each simulation, one step ahead forecasts of y t+1 are formed from the martingale difference null and from rolling estimates of a regression of y t on X t = (1,x t-1 )N. 2. In each experiment, power is calculated by comparing the test statistics against simulation critical values, calculated as the 90th percentile of the distributions of the statistics in the corresponding size experiment reported in Table The number of simulations is 10, See the notes to Table 1.
41 Table 5 Size-Adjusted Power: DGP 2 Empirical Size = 10% A. R=60 P=48 P=96 P=144 P=240 P=480 P=1200 MSPE-adjusted MSPE CCS B. R=120 P=48 P=96 P=144 P=240 P=480 P=1200 MSPE-adjusted MSPE CCS C. R=240 P=48 P=96 P=144 P=240 P=480 P=1200 MSPE-adjusted MSPE CCS Notes: 1. The coefficient b is set to Remaining details are as described in the notes to Table 4.
42 Table 6 Forecasts of Monthly Changes in U.S. Dollar Exchange Rates (1) (2) (3) (4) (5) (6) (7) (8) (9) MSPE- MSPEcountry prediction adjusted normal ^ ^ ^ sample adj. F 2 2 -adj ^F2 1 -(^F 2 2 -adj.) ^F2 1 -^F 2 2 CCS F 2 1 F 2 2 Canada 1990: :10 (0.08) 1.78 ** Japan 1990: :10 (0.43) * Switzerland 1985: :10 (0.48) 1.88 ** U.K. 1985: :10 (0.33) Notes: 1. In column (3), ^F 2 1 is the out of sample MSPE of the no change or random walk model, which forecasts a value of zero for the one month ahead change in the exchange rate. 2. In column (4), ^F 2 2 is the out of sample MSPE of a model that regresses the exchange rate on a constant and the previous month s cross-country interest differentials. The estimated regression vector and the current month s interest differential are then used to predict next month s exchange rate. Rolling regressions are used, with a sample size R of 120 months. 3. In column (5), adj. is the adjustment term P -1 G T t=t-p+1 (X t+1 N ^$ t ) 2, where: P is the number of predictions, P = 166 for Canada and Japan, P=226 for Switzerland and the U.K.; T=2003:10; X t+1 =(constant, interest differential at end of month t)n; ^$ t is the estimated regression vector. 4. In columns (7)-(9), standard errors are in parentheses and t-statistics (columns (7) and (8)) or a P 2 (2) statistic (column (9)) are in italics. Standard errors are computed as described in the text. * and ** denote test statistics significant at the 10 and 5 percent level, respectively, based on one sided tests using critical values from a standard normal (columns (7) and (8)) or chi-squared (CCS) distribution. In columns (7)and (8), and denote statistics significant at the 10 and 5 percent level based on McCracken s (2000) asymptotic critical values. 5. Data are described in the text. See notes to earlier tables for additional definitions.
Approximately normal tests for equal predictive accuracy in nested models
Journal of Econometrics 138 (2007) 291 311 www.elsevier.com/locate/jeconom Approximately normal tests for equal predictive accuracy in nested models Todd E. Clark a,, Kenneth D. West b a Economic Research
More informationStatistical Tests for Multiple Forecast Comparison
Statistical Tests for Multiple Forecast Comparison Roberto S. Mariano (Singapore Management University & University of Pennsylvania) Daniel Preve (Uppsala University) June 6-7, 2008 T.W. Anderson Conference,
More informationOnline Appendices to the Corporate Propensity to Save
Online Appendices to the Corporate Propensity to Save Appendix A: Monte Carlo Experiments In order to allay skepticism of empirical results that have been produced by unusual estimators on fairly small
More informationOut-of-Sample Forecast Tests Robust to the Window Size Choice
Out-of-Sample Forecast Tests Robust to the Window Size Choice Barbara Rossi and Atsushi Inoue Duke University and NC State January 6, 20 Abstract This paper proposes new methodologies for evaluating out-of-sample
More informationPart 2: Analysis of Relationship Between Two Variables
Part 2: Analysis of Relationship Between Two Variables Linear Regression Linear correlation Significance Tests Multiple regression Linear Regression Y = a X + b Dependent Variable Independent Variable
More informationLesson19: Comparing Predictive Accuracy of two Forecasts: Th. Diebold-Mariano Test
Lesson19: Comparing Predictive Accuracy of two Forecasts: The Diebold-Mariano Test Dipartimento di Ingegneria e Scienze dell Informazione e Matematica Università dell Aquila, umberto.triacca@univaq.it
More informationSome useful concepts in univariate time series analysis
Some useful concepts in univariate time series analysis Autoregressive moving average models Autocorrelation functions Model Estimation Diagnostic measure Model selection Forecasting Assumptions: 1. Non-seasonal
More informationPredictability of Non-Linear Trading Rules in the US Stock Market Chong & Lam 2010
Department of Mathematics QF505 Topics in quantitative finance Group Project Report Predictability of on-linear Trading Rules in the US Stock Market Chong & Lam 010 ame: Liu Min Qi Yichen Zhang Fengtian
More informationOverview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model
Overview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model 1 September 004 A. Introduction and assumptions The classical normal linear regression model can be written
More informationThe Changing Relationship Between Commodity Prices and Prices of Other Assets with Global Market Integration
The Changing Relationship Between Commodity Prices and Prices of Other Assets with Global Market Integration Barbara Rossi ICREA, CREI, Duke, UPF, BGSE January 31, 1 Abstract We explore the linkage between
More informationJournal of International Money and Finance
Journal of International Money and Finance 30 (2011) 246 267 Contents lists available at ScienceDirect Journal of International Money and Finance journal homepage: www.elsevier.com/locate/jimf Why panel
More informationFinancial TIme Series Analysis: Part II
Department of Mathematics and Statistics, University of Vaasa, Finland January 29 February 13, 2015 Feb 14, 2015 1 Univariate linear stochastic models: further topics Unobserved component model Signal
More informationTime Series Analysis
Time Series Analysis hm@imm.dtu.dk Informatics and Mathematical Modelling Technical University of Denmark DK-2800 Kgs. Lyngby 1 Outline of the lecture Identification of univariate time series models, cont.:
More informationSYSTEMS OF REGRESSION EQUATIONS
SYSTEMS OF REGRESSION EQUATIONS 1. MULTIPLE EQUATIONS y nt = x nt n + u nt, n = 1,...,N, t = 1,...,T, x nt is 1 k, and n is k 1. This is a version of the standard regression model where the observations
More information2. Linear regression with multiple regressors
2. Linear regression with multiple regressors Aim of this section: Introduction of the multiple regression model OLS estimation in multiple regression Measures-of-fit in multiple regression Assumptions
More informationLeast Squares Estimation
Least Squares Estimation SARA A VAN DE GEER Volume 2, pp 1041 1045 in Encyclopedia of Statistics in Behavioral Science ISBN-13: 978-0-470-86080-9 ISBN-10: 0-470-86080-4 Editors Brian S Everitt & David
More informationInstitute of Actuaries of India Subject CT3 Probability and Mathematical Statistics
Institute of Actuaries of India Subject CT3 Probability and Mathematical Statistics For 2015 Examinations Aim The aim of the Probability and Mathematical Statistics subject is to provide a grounding in
More informationPITFALLS IN TIME SERIES ANALYSIS. Cliff Hurvich Stern School, NYU
PITFALLS IN TIME SERIES ANALYSIS Cliff Hurvich Stern School, NYU The t -Test If x 1,..., x n are independent and identically distributed with mean 0, and n is not too small, then t = x 0 s n has a standard
More informationCHAPTER 13 SIMPLE LINEAR REGRESSION. Opening Example. Simple Regression. Linear Regression
Opening Example CHAPTER 13 SIMPLE LINEAR REGREION SIMPLE LINEAR REGREION! Simple Regression! Linear Regression Simple Regression Definition A regression model is a mathematical equation that descries the
More informationRegression Analysis: A Complete Example
Regression Analysis: A Complete Example This section works out an example that includes all the topics we have discussed so far in this chapter. A complete example of regression analysis. PhotoDisc, Inc./Getty
More informationA Century of PPP: Supportive Results from non-linear Unit Root Tests
Mohsen Bahmani-Oskooee a, Ali Kutan b and Su Zhou c A Century of PPP: Supportive Results from non-linear Unit Root Tests ABSTRACT Testing for stationarity of the real exchange rates is a common practice
More informationThe Canadian dollar as a commodity currency: Has the relationship changed?* Monica Mow Supervisor: Dr. Graham Voss
1 The Canadian dollar as a commodity currency: Has the relationship changed?* By Monica Mow Supervisor: Dr. Graham Voss A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of BACHELOR
More informationTime Series Analysis
Time Series Analysis Identifying possible ARIMA models Andrés M. Alonso Carolina García-Martos Universidad Carlos III de Madrid Universidad Politécnica de Madrid June July, 2012 Alonso and García-Martos
More informationINFLATION, INTEREST RATE, AND EXCHANGE RATE: WHAT IS THE RELATIONSHIP?
107 INFLATION, INTEREST RATE, AND EXCHANGE RATE: WHAT IS THE RELATIONSHIP? Maurice K. Shalishali, Columbus State University Johnny C. Ho, Columbus State University ABSTRACT A test of IFE (International
More informationIs the Forward Exchange Rate a Useful Indicator of the Future Exchange Rate?
Is the Forward Exchange Rate a Useful Indicator of the Future Exchange Rate? Emily Polito, Trinity College In the past two decades, there have been many empirical studies both in support of and opposing
More informationPlease follow the directions once you locate the Stata software in your computer. Room 114 (Business Lab) has computers with Stata software
STATA Tutorial Professor Erdinç Please follow the directions once you locate the Stata software in your computer. Room 114 (Business Lab) has computers with Stata software 1.Wald Test Wald Test is used
More informationTHE UNIVERSITY OF CHICAGO, Booth School of Business Business 41202, Spring Quarter 2014, Mr. Ruey S. Tsay. Solutions to Homework Assignment #2
THE UNIVERSITY OF CHICAGO, Booth School of Business Business 41202, Spring Quarter 2014, Mr. Ruey S. Tsay Solutions to Homework Assignment #2 Assignment: 1. Consumer Sentiment of the University of Michigan.
More informationDescriptive Statistics
Descriptive Statistics Primer Descriptive statistics Central tendency Variation Relative position Relationships Calculating descriptive statistics Descriptive Statistics Purpose to describe or summarize
More informationHow To Understand The Theory Of Probability
Graduate Programs in Statistics Course Titles STAT 100 CALCULUS AND MATR IX ALGEBRA FOR STATISTICS. Differential and integral calculus; infinite series; matrix algebra STAT 195 INTRODUCTION TO MATHEMATICAL
More informationResearch Division Federal Reserve Bank of St. Louis Working Paper Series
Research Division Federal Reserve Bank of St. Louis Working Paper Series Comment on "Taylor Rule Exchange Rate Forecasting During the Financial Crisis" Michael W. McCracken Working Paper 2012-030A http://research.stlouisfed.org/wp/2012/2012-030.pdf
More informationVector Time Series Model Representations and Analysis with XploRe
0-1 Vector Time Series Model Representations and Analysis with plore Julius Mungo CASE - Center for Applied Statistics and Economics Humboldt-Universität zu Berlin mungo@wiwi.hu-berlin.de plore MulTi Motivation
More informationWORKING PAPER NO. 11-31 OUT-OF-SAMPLE FORECAST TESTS ROBUST TO THE CHOICE OF WINDOW SIZE
WORKING PAPER NO. 11-31 OUT-OF-SAMPLE FORECAST TESTS ROBUST TO THE CHOICE OF WINDOW SIZE Barbara Rossi Duke University and Visiting Scholar, Federal Reserve Bank of Philadelphia Atsushi Inoue North Carolina
More informationThe VAR models discussed so fare are appropriate for modeling I(0) data, like asset returns or growth rates of macroeconomic time series.
Cointegration The VAR models discussed so fare are appropriate for modeling I(0) data, like asset returns or growth rates of macroeconomic time series. Economic theory, however, often implies equilibrium
More informationStatistics Graduate Courses
Statistics Graduate Courses STAT 7002--Topics in Statistics-Biological/Physical/Mathematics (cr.arr.).organized study of selected topics. Subjects and earnable credit may vary from semester to semester.
More informationNon-Stationary Time Series andunitroottests
Econometrics 2 Fall 2005 Non-Stationary Time Series andunitroottests Heino Bohn Nielsen 1of25 Introduction Many economic time series are trending. Important to distinguish between two important cases:
More informationSimple Linear Regression Inference
Simple Linear Regression Inference 1 Inference requirements The Normality assumption of the stochastic term e is needed for inference even if it is not a OLS requirement. Therefore we have: Interpretation
More informationCurriculum Map Statistics and Probability Honors (348) Saugus High School Saugus Public Schools 2009-2010
Curriculum Map Statistics and Probability Honors (348) Saugus High School Saugus Public Schools 2009-2010 Week 1 Week 2 14.0 Students organize and describe distributions of data by using a number of different
More informationCorporate Defaults and Large Macroeconomic Shocks
Corporate Defaults and Large Macroeconomic Shocks Mathias Drehmann Bank of England Andrew Patton London School of Economics and Bank of England Steffen Sorensen Bank of England The presentation expresses
More informationChapter 1. Vector autoregressions. 1.1 VARs and the identi cation problem
Chapter Vector autoregressions We begin by taking a look at the data of macroeconomics. A way to summarize the dynamics of macroeconomic data is to make use of vector autoregressions. VAR models have become
More informationChapter 6: Multivariate Cointegration Analysis
Chapter 6: Multivariate Cointegration Analysis 1 Contents: Lehrstuhl für Department Empirische of Wirtschaftsforschung Empirical Research and und Econometrics Ökonometrie VI. Multivariate Cointegration
More informationAre Macroeconomic Variables Useful for Forecasting the Distribution of U.S. Inflation?
Are Macroeconomic Variables Useful for Forecasting the Distribution of U.S. Inflation? Sebastiano Manzan a and Dawit Zerom b a Department of Economics & Finance, Baruch College, CUNY b California State
More informationPractical. I conometrics. data collection, analysis, and application. Christiana E. Hilmer. Michael J. Hilmer San Diego State University
Practical I conometrics data collection, analysis, and application Christiana E. Hilmer Michael J. Hilmer San Diego State University Mi Table of Contents PART ONE THE BASICS 1 Chapter 1 An Introduction
More information1 Short Introduction to Time Series
ECONOMICS 7344, Spring 202 Bent E. Sørensen January 24, 202 Short Introduction to Time Series A time series is a collection of stochastic variables x,.., x t,.., x T indexed by an integer value t. The
More informationThe Phillips Curve in an Era. of Well-Anchored Inflation Expectations
FEDERAL RESERVE BANK OF SAN FRANCISCO UNPUBLISHED WORKING PAPER The Phillips Curve in an Era of Well-Anchored Inflation Expectations John C. Williams Federal Reserve Bank of San Francisco September 26
More informationFactors affecting online sales
Factors affecting online sales Table of contents Summary... 1 Research questions... 1 The dataset... 2 Descriptive statistics: The exploratory stage... 3 Confidence intervals... 4 Hypothesis tests... 4
More informationEmpirical Properties of the Indonesian Rupiah: Testing for Structural Breaks, Unit Roots, and White Noise
Volume 24, Number 2, December 1999 Empirical Properties of the Indonesian Rupiah: Testing for Structural Breaks, Unit Roots, and White Noise Reza Yamora Siregar * 1 This paper shows that the real exchange
More information1) Write the following as an algebraic expression using x as the variable: Triple a number subtracted from the number
1) Write the following as an algebraic expression using x as the variable: Triple a number subtracted from the number A. 3(x - x) B. x 3 x C. 3x - x D. x - 3x 2) Write the following as an algebraic expression
More informationIs the Basis of the Stock Index Futures Markets Nonlinear?
University of Wollongong Research Online Applied Statistics Education and Research Collaboration (ASEARC) - Conference Papers Faculty of Engineering and Information Sciences 2011 Is the Basis of the Stock
More informationChapter 9: Univariate Time Series Analysis
Chapter 9: Univariate Time Series Analysis In the last chapter we discussed models with only lags of explanatory variables. These can be misleading if: 1. The dependent variable Y t depends on lags of
More informationWeek TSX Index 1 8480 2 8470 3 8475 4 8510 5 8500 6 8480
1) The S & P/TSX Composite Index is based on common stock prices of a group of Canadian stocks. The weekly close level of the TSX for 6 weeks are shown: Week TSX Index 1 8480 2 8470 3 8475 4 8510 5 8500
More information1. What is the critical value for this 95% confidence interval? CV = z.025 = invnorm(0.025) = 1.96
1 Final Review 2 Review 2.1 CI 1-propZint Scenario 1 A TV manufacturer claims in its warranty brochure that in the past not more than 10 percent of its TV sets needed any repair during the first two years
More informationTURUN YLIOPISTO UNIVERSITY OF TURKU TALOUSTIEDE DEPARTMENT OF ECONOMICS RESEARCH REPORTS. A nonlinear moving average test as a robust test for ARCH
TURUN YLIOPISTO UNIVERSITY OF TURKU TALOUSTIEDE DEPARTMENT OF ECONOMICS RESEARCH REPORTS ISSN 0786 656 ISBN 951 9 1450 6 A nonlinear moving average test as a robust test for ARCH Jussi Tolvi No 81 May
More informationTesting for Granger causality between stock prices and economic growth
MPRA Munich Personal RePEc Archive Testing for Granger causality between stock prices and economic growth Pasquale Foresti 2006 Online at http://mpra.ub.uni-muenchen.de/2962/ MPRA Paper No. 2962, posted
More informationChapter 5. Analysis of Multiple Time Series. 5.1 Vector Autoregressions
Chapter 5 Analysis of Multiple Time Series Note: The primary references for these notes are chapters 5 and 6 in Enders (2004). An alternative, but more technical treatment can be found in chapters 10-11
More informationSales forecasting # 2
Sales forecasting # 2 Arthur Charpentier arthur.charpentier@univ-rennes1.fr 1 Agenda Qualitative and quantitative methods, a very general introduction Series decomposition Short versus long term forecasting
More informationTime Series Analysis in Economics. Klaus Neusser
Time Series Analysis in Economics Klaus Neusser May 26, 2015 Contents I Univariate Time Series Analysis 3 1 Introduction 1 1.1 Some examples.......................... 2 1.2 Formal definitions.........................
More informationStandard errors of marginal effects in the heteroskedastic probit model
Standard errors of marginal effects in the heteroskedastic probit model Thomas Cornelißen Discussion Paper No. 320 August 2005 ISSN: 0949 9962 Abstract In non-linear regression models, such as the heteroskedastic
More informationOut-of-Sample Forecast Tests Robust to the Choice of Window Size
Out-of-Sample Forecast Tests Robust to the Choice of Window Size Barbara Rossi and Atsushi Inoue (ICREA,UPF,CREI,BGSE,Duke) (NC State) April 1, 2012 Abstract This paper proposes new methodologies for evaluating
More informationPerforming Unit Root Tests in EViews. Unit Root Testing
Página 1 de 12 Unit Root Testing The theory behind ARMA estimation is based on stationary time series. A series is said to be (weakly or covariance) stationary if the mean and autocovariances of the series
More informationMULTIPLE REGRESSIONS ON SOME SELECTED MACROECONOMIC VARIABLES ON STOCK MARKET RETURNS FROM 1986-2010
Advances in Economics and International Finance AEIF Vol. 1(1), pp. 1-11, December 2014 Available online at http://www.academiaresearch.org Copyright 2014 Academia Research Full Length Research Paper MULTIPLE
More informationMonitoring Structural Change in Dynamic Econometric Models
Monitoring Structural Change in Dynamic Econometric Models Achim Zeileis Friedrich Leisch Christian Kleiber Kurt Hornik http://www.ci.tuwien.ac.at/~zeileis/ Contents Model frame Generalized fluctuation
More informationFrom the help desk: Bootstrapped standard errors
The Stata Journal (2003) 3, Number 1, pp. 71 80 From the help desk: Bootstrapped standard errors Weihua Guan Stata Corporation Abstract. Bootstrapping is a nonparametric approach for evaluating the distribution
More informationForecasting Chilean Industrial Production and Sales with Automated Procedures 1
Forecasting Chilean Industrial Production and Sales with Automated Procedures 1 Rómulo A. Chumacero 2 February 2004 1 I thank Ernesto Pastén, Klaus Schmidt-Hebbel, and Rodrigo Valdés for helpful comments
More informationCONTROL SYSTEMS, ROBOTICS, AND AUTOMATION - Vol. V - Relations Between Time Domain and Frequency Domain Prediction Error Methods - Tomas McKelvey
COTROL SYSTEMS, ROBOTICS, AD AUTOMATIO - Vol. V - Relations Between Time Domain and Frequency Domain RELATIOS BETWEE TIME DOMAI AD FREQUECY DOMAI PREDICTIO ERROR METHODS Tomas McKelvey Signal Processing,
More informationService courses for graduate students in degree programs other than the MS or PhD programs in Biostatistics.
Course Catalog In order to be assured that all prerequisites are met, students must acquire a permission number from the education coordinator prior to enrolling in any Biostatistics course. Courses are
More informationFinal Exam Practice Problem Answers
Final Exam Practice Problem Answers The following data set consists of data gathered from 77 popular breakfast cereals. The variables in the data set are as follows: Brand: The brand name of the cereal
More informationOnline Appendix. Supplemental Material for Insider Trading, Stochastic Liquidity and. Equilibrium Prices. by Pierre Collin-Dufresne and Vyacheslav Fos
Online Appendix Supplemental Material for Insider Trading, Stochastic Liquidity and Equilibrium Prices by Pierre Collin-Dufresne and Vyacheslav Fos 1. Deterministic growth rate of noise trader volatility
More informationDaily Changes in Fed Funds Futures Prices
Daily Changes in Fed Funds Futures Prices James D. Hamilton, Professor of Economics University of California, San Diego jhamilton@ucsd.edu December 19, 2006 Revised: July 6, 2007 JEL codes: E40, E50, G13
More informationLean Six Sigma Analyze Phase Introduction. TECH 50800 QUALITY and PRODUCTIVITY in INDUSTRY and TECHNOLOGY
TECH 50800 QUALITY and PRODUCTIVITY in INDUSTRY and TECHNOLOGY Before we begin: Turn on the sound on your computer. There is audio to accompany this presentation. Audio will accompany most of the online
More informationUnivariate and Multivariate Methods PEARSON. Addison Wesley
Time Series Analysis Univariate and Multivariate Methods SECOND EDITION William W. S. Wei Department of Statistics The Fox School of Business and Management Temple University PEARSON Addison Wesley Boston
More informationDepartment of Economics
Department of Economics Working Paper Do Stock Market Risk Premium Respond to Consumer Confidence? By Abdur Chowdhury Working Paper 2011 06 College of Business Administration Do Stock Market Risk Premium
More informationTechnical Efficiency Accounting for Environmental Influence in the Japanese Gas Market
Technical Efficiency Accounting for Environmental Influence in the Japanese Gas Market Sumiko Asai Otsuma Women s University 2-7-1, Karakida, Tama City, Tokyo, 26-854, Japan asai@otsuma.ac.jp Abstract:
More informationFULLY MODIFIED OLS FOR HETEROGENEOUS COINTEGRATED PANELS
FULLY MODIFIED OLS FOR HEEROGENEOUS COINEGRAED PANELS Peter Pedroni ABSRAC his chapter uses fully modified OLS principles to develop new methods for estimating and testing hypotheses for cointegrating
More informationAppendices with Supplementary Materials for CAPM for Estimating Cost of Equity Capital: Interpreting the Empirical Evidence
Appendices with Supplementary Materials for CAPM for Estimating Cost of Equity Capital: Interpreting the Empirical Evidence This document contains supplementary material to the paper titled CAPM for estimating
More informationFinance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C.
Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Government Debt and Macroeconomic Activity: A Predictive Analysis
More informationOn Marginal Effects in Semiparametric Censored Regression Models
On Marginal Effects in Semiparametric Censored Regression Models Bo E. Honoré September 3, 2008 Introduction It is often argued that estimation of semiparametric censored regression models such as the
More informationbusiness statistics using Excel OXFORD UNIVERSITY PRESS Glyn Davis & Branko Pecar
business statistics using Excel Glyn Davis & Branko Pecar OXFORD UNIVERSITY PRESS Detailed contents Introduction to Microsoft Excel 2003 Overview Learning Objectives 1.1 Introduction to Microsoft Excel
More informationB.3. Robustness: alternative betas estimation
Appendix B. Additional empirical results and robustness tests This Appendix contains additional empirical results and robustness tests. B.1. Sharpe ratios of beta-sorted portfolios Fig. B1 plots the Sharpe
More informationITSM-R Reference Manual
ITSM-R Reference Manual George Weigt June 5, 2015 1 Contents 1 Introduction 3 1.1 Time series analysis in a nutshell............................... 3 1.2 White Noise Variance.....................................
More informationStatistics I for QBIC. Contents and Objectives. Chapters 1 7. Revised: August 2013
Statistics I for QBIC Text Book: Biostatistics, 10 th edition, by Daniel & Cross Contents and Objectives Chapters 1 7 Revised: August 2013 Chapter 1: Nature of Statistics (sections 1.1-1.6) Objectives
More informationNon Parametric Inference
Maura Department of Economics and Finance Università Tor Vergata Outline 1 2 3 Inverse distribution function Theorem: Let U be a uniform random variable on (0, 1). Let X be a continuous random variable
More informationInternet Appendix to Stock Market Liquidity and the Business Cycle
Internet Appendix to Stock Market Liquidity and the Business Cycle Randi Næs, Johannes A. Skjeltorp and Bernt Arne Ødegaard This Internet appendix contains additional material to the paper Stock Market
More informationWhat s New in Econometrics? Lecture 8 Cluster and Stratified Sampling
What s New in Econometrics? Lecture 8 Cluster and Stratified Sampling Jeff Wooldridge NBER Summer Institute, 2007 1. The Linear Model with Cluster Effects 2. Estimation with a Small Number of Groups and
More informationTime Series Analysis
Time Series Analysis Forecasting with ARIMA models Andrés M. Alonso Carolina García-Martos Universidad Carlos III de Madrid Universidad Politécnica de Madrid June July, 2012 Alonso and García-Martos (UC3M-UPM)
More informationECON 142 SKETCH OF SOLUTIONS FOR APPLIED EXERCISE #2
University of California, Berkeley Prof. Ken Chay Department of Economics Fall Semester, 005 ECON 14 SKETCH OF SOLUTIONS FOR APPLIED EXERCISE # Question 1: a. Below are the scatter plots of hourly wages
More informationRobust Inferences from Random Clustered Samples: Applications Using Data from the Panel Survey of Income Dynamics
Robust Inferences from Random Clustered Samples: Applications Using Data from the Panel Survey of Income Dynamics John Pepper Assistant Professor Department of Economics University of Virginia 114 Rouss
More informationII. DISTRIBUTIONS distribution normal distribution. standard scores
Appendix D Basic Measurement And Statistics The following information was developed by Steven Rothke, PhD, Department of Psychology, Rehabilitation Institute of Chicago (RIC) and expanded by Mary F. Schmidt,
More informationCentre for Central Banking Studies
Centre for Central Banking Studies Technical Handbook No. 4 Applied Bayesian econometrics for central bankers Andrew Blake and Haroon Mumtaz CCBS Technical Handbook No. 4 Applied Bayesian econometrics
More informationClass 19: Two Way Tables, Conditional Distributions, Chi-Square (Text: Sections 2.5; 9.1)
Spring 204 Class 9: Two Way Tables, Conditional Distributions, Chi-Square (Text: Sections 2.5; 9.) Big Picture: More than Two Samples In Chapter 7: We looked at quantitative variables and compared the
More informationMULTIPLE REGRESSION AND ISSUES IN REGRESSION ANALYSIS
MULTIPLE REGRESSION AND ISSUES IN REGRESSION ANALYSIS MSR = Mean Regression Sum of Squares MSE = Mean Squared Error RSS = Regression Sum of Squares SSE = Sum of Squared Errors/Residuals α = Level of Significance
More informationFACTOR MODEL FORECASTS OF EXCHANGE RATES. Charles Engel University of Wisconsin. Nelson C. Mark University of Notre Dame
FACTOR MODEL FORECASTS OF EXCHANGE RATES Charles Engel University of Wisconsin Nelson C. Mark University of Notre Dame Kenneth D. West University of Wisconsin November 2008 Revised November 2009 ABSTRACT
More informationTesting against a Change from Short to Long Memory
Testing against a Change from Short to Long Memory Uwe Hassler and Jan Scheithauer Goethe-University Frankfurt This version: December 9, 2007 Abstract This paper studies some well-known tests for the null
More informationVICENTIU COVRIG* BUEN SIN LOW
THE QUALITY OF VOLATILITY TRADED ON THE OVER-THE-COUNTER CURRENCY MARKET: A MULTIPLE HORIZONS STUDY VICENTIU COVRIG* BUEN SIN LOW Previous studies of the quality of market-forecasted volatility have used
More informationAssociation Between Variables
Contents 11 Association Between Variables 767 11.1 Introduction............................ 767 11.1.1 Measure of Association................. 768 11.1.2 Chapter Summary.................... 769 11.2 Chi
More information" Y. Notation and Equations for Regression Lecture 11/4. Notation:
Notation: Notation and Equations for Regression Lecture 11/4 m: The number of predictor variables in a regression Xi: One of multiple predictor variables. The subscript i represents any number from 1 through
More informationIntroduction to General and Generalized Linear Models
Introduction to General and Generalized Linear Models General Linear Models - part I Henrik Madsen Poul Thyregod Informatics and Mathematical Modelling Technical University of Denmark DK-2800 Kgs. Lyngby
More informationMaster programme in Statistics
Master programme in Statistics Björn Holmquist 1 1 Department of Statistics Lund University Cramérsällskapets årskonferens, 2010-03-25 Master programme Vad är ett Master programme? Breddmaster vs Djupmaster
More informationSection Format Day Begin End Building Rm# Instructor. 001 Lecture Tue 6:45 PM 8:40 PM Silver 401 Ballerini
NEW YORK UNIVERSITY ROBERT F. WAGNER GRADUATE SCHOOL OF PUBLIC SERVICE Course Syllabus Spring 2016 Statistical Methods for Public, Nonprofit, and Health Management Section Format Day Begin End Building
More informationIntroduction to Regression and Data Analysis
Statlab Workshop Introduction to Regression and Data Analysis with Dan Campbell and Sherlock Campbell October 28, 2008 I. The basics A. Types of variables Your variables may take several forms, and it
More informationPearson s Correlation
Pearson s Correlation Correlation the degree to which two variables are associated (co-vary). Covariance may be either positive or negative. Its magnitude depends on the units of measurement. Assumes the
More information