# Chapter 4: Vector Autoregressive Models

Size: px
Start display at page:

## Transcription

1 Chapter 4: Vector Autoregressive Models 1

2 Contents: Lehrstuhl für Department Empirische of Wirtschaftsforschung Empirical Research and und Econometrics Ökonometrie IV.1 Vector Autoregressive Models (VAR)... 3 IV.1.1 Introduction... 3 IV.1.2 Modelling of a VAR(1)-Models... 5 IV.1.3 VAR(p)-Models with more than two Variables... 8 IV.1.4 Estimation of VAR-Models... 9 IV.2 Granger Causality Test and Blockexogeneity IV.3 Impulse Response IV.4 Orthogonal Impulse Response IV.5 Variance Decomposition

3 IV.1 Vector Autoregressive Models (VAR) IV.1.1 Introduction In ARIMA models we only derive the actual value from past values for an endogenous variable. However, there is often no theoretical background available. Therefore, we can use Vector Autoregressive (VAR) Models. 1. The single equation approach explains an endogenous variable (e.g. private consumption) by a range of other variables (e.g., disposal income, property, interest rate) Assumption: Explanatory variables are exogenous. But: Macroeconomic variables are often not exogenous (endogenity problem). 2. In theory exists no assumptions about the dynamic adjustment. Dynamic modelling of the consumption function, e.g. adjustment of consumption behaviour to substantial income, occurs only slowly. 3

4 Multivariarte (linear) time series models eliminate both problems. Development of a variable is explained by the development of potential explanatory variables. Value is explained by its own history and simultaneously by considering various variables and their history. e.g Dependency of the long-term interest rate from the short run interest rate: Is difficult in univariate time series analysis, because one would need estimations of the development of short run interest rate. Therefore, it is not possible to make good forecast out of such a model. In contrast to a bivariate model, where both, the history of long and short term interest rate are taken into consideration. With vector autoregressive models it is possible to approximate the actual process by arbitrarily choosing lagged variables. Thereby, one can form economic variables into a time series model without an explicit theoretical idea of the dynamic relations. 4

5 IV.1.2 Modelling of a VAR(1)-Models The most easy multivariate time series model is the bivariate vector autoregressive model with two dependent variables y 1,t and y 2,t, where t = 1,..., T. The development of the series should be explained by the common past of these variables. That means, the explanatory variables in the simplest model are y 1,t-1 and y 2,t-1. The VAR(1) with lagged values for every variable is determined by: Matrix Notation: y, α y, α y, ε, y, α y, α y, ε, y A y ε A α α α α 5

6 Assumptions about the Error Terms: 1. The expected residuals are zero: E ε, 0 with i = 1, 2 2. The error terms are not autocorrelated: E ε, ε, 0 with t τ 6

7 Interpretation of VAR Models: VAR-Models themselves do not allow us to make statements about causal relationships. This holds especially when VAR-Models are only approximately adjusted to an unknown time series process, while a causal interpretation requires an underlying economic model. However, VAR-Models allow interpretations about the dynamic relationship between the indicated variables. Granger Causality Be careful, Granger Causality is a much weaker argument than normal causality. If y 2 has any influence on y 1 (α 0), one can say y 2 Granger causes y 1. On the other hand, if y 1 has any influence on y 2 (α 0), one can say y 1 Granger causes y 2. 7

8 IV.1.3 VAR(p)-Models with more than two Variables An VAR(p)-Model, with p variables, is given as: y A y A y A y ε If one wants to expand the equation with a trend, intercept or seasonal adjustment, it will be necessary to augment the Vector xt, which includes all the deterministic components, and the matrix B (VARX-Model): y A y A y A y Bx ε 8

9 IV.1.4 Estimation of VAR-Models Specifications: 1. Determination of endogenous variables according to economic theory, empirical evidence and experience. 2. Transformation of time series (take logs or log-returns). 3. Insert seasonal component, especially for macro data. 4. Control for deterministic terms. Estimation according to LS-Regression If the error terms between the variables are uncorrelated, the estimation will be unbiased and efficient. 9

10 Determination of Lag Length: The determination of lag length is a trade-off between the curse of dimensionality and abbreviate models, which are not appropriate to indicate the dynamic adjustment. If the lag length is to short, autocorrelation of the error terms could lead to apparently significant and inefficient estimators. Therefore, one would receive wrong results. With the so called curse of dimensionality we understand, that even with a relatively small lag length a large number of parameters if required. On the other hand, with increasing number of parameters, the degrees of freedom decrease, which could possibly result in significant of inefficient estimators. Information Criteria: The idea of information criteria is similar to the trade-off discussed above. On the one hand, the model should be able to reflect the observed process as precise as possible (error terms should be as small as possible) and on the other hand, to many variables lead to inefficient estimators. Therefore, the information criteria are combined out of the squared sum of residuals and a penalty 10

11 term for the number of lags. In detail, for T observations we chose the lag length p in a way that the reduction of the squared residuals after augmenting lag p+1, is smaller than the according boost in the penalty term. Squared residuals: Penalty terms: ln AIC: SIC: HQ: with constant c > 1 11

12 Statistical Tests: Lehrstuhl für Department Empirische of Wirtschaftsforschung Empirical Research and und Econometrics Ökonometrie An alternative to the global preparation are local statistic tests. For example, one could use Log Likelihood-Ratio Test (LR) to specify the lag length. Based on a VAR-Model with lag length p, one can check if the explanatory power of the model increased after taken lag p 1 into consideration. With the support of the LR test, one can control if the change in the power is significant or if it has only a power comparable to a random variable. Formally the LR Test is defined as, λ L β L β where r stands for restricted and ur for unrestricted model. Hence, it is tested whether it is possible to restrict all p+1 lags to zero. Additional lags cannot increase the power of estimation if λ is close to one, but if λ is close to zero. Therefore, the usual Wald-Test, which is adequate for an -distribution can be used. Unfortunately, the LR-test is not sufficient for models with missing lag structures. 12

13 IV.2 Granger Causality Test and Blockexogeneity Granger (1969) developed a test approach to proof if a time series X contribute to the prediction of another series Y. Granger Causality is exists if the mean squared forecast error (MSE) by using the series X in the forecast model is smaller than without consideration of X: MSE (Ŷ I ) < MSE (Ŷ (I without X, s = Y t+ h t Y t+ h t t s 1,...,n)) for at least one h = 1, 2,, m with: MSE = 1 N N i= 1 (Yi Ŷi ) 2 and forecasts, h equals the forecast horizon. Ŷ i equals the forecast in time point i and N equals the number of 13

14 In the bivariate approach, it follows: t n m = α + β Y + δ X i t i i t j i= 1 j= 1 Y + ε t The test statistic of the Wald test (W) equals to J F. Thereby, J is the number of restrictions to test (in the above case J = m). F denotes the value of the F statistic with F = ' ' (ˆ ε ˆ ˆ rεr ε εˆ) / J ' εˆ εˆ /(T K) where ˆ ' ε r ε equals the sum of the squared residuals by imposition of restrictions (δ 1 = δ 2 = = δ m = 0) rˆ and εˆ ' εˆ is the sum of squared residuals of the estimation without restrictions. 14

15 T is the number of observations and K the number of regressors of the model. The test statistic W is asymptotically χ 2 -distributed with J degrees of freedom. Test for Blockexogeneity: Hypothesis: - y 3 is (block) exogenous for y 1 and y 2 H : α α 0 - y 2 and y 3 are (block) exogenous for y 1 H : α α 0 y, α y, α y, α y, ε, y, α y, α y, α y, ε, y, α y, α y, α y, ε, 15

16 IV.3 Impulse Response The dynamic adjustment of reciprocal dependency is immediately not considerable. The impulse response test shows the effects of an exogenous shock on the whole process over time. Therefore, one can detect the dynamic relationships over time. Idea: Initially, look at the adjustment of the endogenous variables over time, after a hypothetical shock in t. This adjustment is compared with the time series process without a shock, i.e. the actual process. The impulse response sequences plot the difference between this two time paths. 16

17 To illustrate this, we assume a two dimensional VAR(1)-Model: y, α y, α y, ε, y, α y, α y, ε, Initially, in t 1 we assume a shock in the error term ε, of the first equation. This shock has a direct effect on y, of exactly the same amount. Whereas y, is not effected, assuming that ε, 0 with t 1,... T. In the second Period (t = 2), the original shock has still an effect over the lagged value of y. The effect on y, is α ε, and the effect on y, is α ε,. In the third period the effect on y, is not only α α ε,, but also α α ε,. Accordingly, the effect on y, is α α ε, α α ε,. Thus, it is possible to obsess the effect of a non-recurring shock in one variable, to all variables over time. One could summarise the result in: y C ε with C I (Vector-Moving-Average Process) and where C are the weight of past stocks. 17

18 Figure 1: Impulse-Response Sequences for Industrial Production and Orders Received 18

19 Figure 1 shows the adjustment of the impulse response sequence for the example of industrial production and orders received, based on a shock in the amount of the standard errors in both variables. On the left side of the figure are plotted the reactions on a shock in industrial production, on the right side the reactions on a shock in orders received. According to our assumption, the industrial production has no effect on orders received and the orders received have no influence on industrial production in the first period. Thus, these graphs start at the point of origin. In contrast, a shock in one variable has an instant effect on its present value. Therefore, the upper-left and lowerright graph begin at the respective standard error. The effects in the following periods depend on the dimension of the coefficients. If the sum of all coefficients in one equation is smaller than one, the effects will decrease over time and will revert to a value close to zero after a certain period. 19

20 IV.4 Orthogonal Impulse Response In the previous impulse response model we assumed that the error terms of the different equation are uncorrelated. However, this assumption is quite restricted. A hypothetical shock in only one equation does not respond a realistic adjustment process. To control for correlation between error terms we have to use the orthogonal impulse response sequences. The idea is to modify the original moving-average construction in a way that the residuals are uncorrelated, i.e. the residuals have to be orthogonal to each other. Therefore, we can write y C ν with C C G, where G is a transformation matrix with the property G Ω G I (Cholesky- Decomposition). 20

21 The error terms of the modified system are ν G ε. The variance-covariance matrix of ν is diagonal, according to the properties of G. However, the G matrix is not clearly defined by the Cholesky decomposition (Ω G G, where Ω is the original variance-covariance matrix). Moreover, we have to specify the order of the variables. The chosen order presumes the causal relationship between the variables. The results of the impulse response can depend highly on the order of the variables, especially when they are highly correlated. Figure 2 shows the orthogonal impulse response sequence for the variable order industrial production, orders received and vice versa. For the direction industrial production and orders received the joint component of error terms are only assigned to industrial production. The impulse response and the orthogonal impulse response with respect to a shock in industrial production are therefore the same. However, the effects on a change in orders received are different. Due to the high correlation, parts of the effect can be assigned to industrial production immediately. As long as the economic theory gives no explicit information about the order of the causal relationships between variables, there is no unique solution. 21

22 Figure 2: Two-Sided Orthogonal Impulse Response for Industrial Production and Orders Received 22

23 IV.5 Variance Decomposition An alternative of impulse response, to receive a compact overview of the dynamic structures of a VAR Model, are variance decomposition sequences. This method is also based on a vector moving average model and orthogonal error terms. In contrast to impulse response, the task of variance decomposition is to achieve information about the forecast ability. The idea is, that even a perfect model involves ambiguity about the realisation of y,, because the error terms associate uncertainty. According to the interactions between the equations, the uncertainty is transformed to all equations. The aim of the decomposition is to reduce the uncertainty in one equation to the variance of error terms in all equations. 23

24 Assume a two-dimensional VAR model (Moving average representation): y, c ν, c ν, c ν, c ν, y, c ν, c ν, c ν, c ν, As the development of the lagged error terms is already known, it exists only uncertainty about the present error terms ν, and ν,. A compact illustration is shown in figure 3, where the share of the single error terms is plotted against time. For the industrial production it is shown that it depends mainly on its own error terms, independent of the order of the variables. However, after a while the orders received gets more and more important and outdistance the industrial production in period 17 or 6, respectively. On the other hand, the variance decomposition of the orders received replies a clear picture. The own variance dominates the variance of industrial production even in the long run. 24

25 Figure 3: Variance Decomposition of Industrial Production and Orders Received 25

### Chapter 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

### Chapter 5: Bivariate Cointegration Analysis

Chapter 5: Bivariate Cointegration Analysis 1 Contents: Lehrstuhl für Department Empirische of Wirtschaftsforschung Empirical Research and und Econometrics Ökonometrie V. Bivariate Cointegration Analysis...

### 16 : Demand Forecasting

16 : Demand Forecasting 1 Session Outline Demand Forecasting Subjective methods can be used only when past data is not available. When past data is available, it is advisable that firms should use statistical

### Chapter 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

### A Trading Strategy Based on the Lead-Lag Relationship of Spot and Futures Prices of the S&P 500

A Trading Strategy Based on the Lead-Lag Relationship of Spot and Futures Prices of the S&P 500 FE8827 Quantitative Trading Strategies 2010/11 Mini-Term 5 Nanyang Technological University Submitted By:

### SYSTEMS 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

### y t by left multiplication with 1 (L) as y t = 1 (L) t =ª(L) t 2.5 Variance decomposition and innovation accounting Consider the VAR(p) model where

. Variance decomposition and innovation accounting Consider the VAR(p) model where (L)y t = t, (L) =I m L L p L p is the lag polynomial of order p with m m coe±cient matrices i, i =,...p. Provided that

### Examining the Relationship between ETFS and Their Underlying Assets in Indian Capital Market

2012 2nd International Conference on Computer and Software Modeling (ICCSM 2012) IPCSIT vol. 54 (2012) (2012) IACSIT Press, Singapore DOI: 10.7763/IPCSIT.2012.V54.20 Examining the Relationship between

### State Space Time Series Analysis

State Space Time Series Analysis p. 1 State Space Time Series Analysis Siem Jan Koopman http://staff.feweb.vu.nl/koopman Department of Econometrics VU University Amsterdam Tinbergen Institute 2011 State

### Sales 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

### An Introduction into the SVAR Methodology: Identification, Interpretation and Limitations of SVAR models

Kiel Institute of World Economics Duesternbrooker Weg 120 24105 Kiel (Germany) Kiel Working Paper No. 1072 An Introduction into the SVAR Methodology: Identification, Interpretation and Limitations of SVAR

### PITFALLS 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

### TIME SERIES ANALYSIS

TIME SERIES ANALYSIS Ramasubramanian V. I.A.S.R.I., Library Avenue, New Delhi- 110 012 ram_stat@yahoo.co.in 1. Introduction A Time Series (TS) is a sequence of observations ordered in time. Mostly these

### ADVANCED FORECASTING MODELS USING SAS SOFTWARE

ADVANCED FORECASTING MODELS USING SAS SOFTWARE Girish Kumar Jha IARI, Pusa, New Delhi 110 012 gjha_eco@iari.res.in 1. Transfer Function Model Univariate ARIMA models are useful for analysis and forecasting

### 4. Simple regression. QBUS6840 Predictive Analytics. https://www.otexts.org/fpp/4

4. Simple regression QBUS6840 Predictive Analytics https://www.otexts.org/fpp/4 Outline The simple linear model Least squares estimation Forecasting with regression Non-linear functional forms Regression

### Vector 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

### Marketing Mix Modelling and Big Data P. M Cain

1) Introduction Marketing Mix Modelling and Big Data P. M Cain Big data is generally defined in terms of the volume and variety of structured and unstructured information. Whereas structured data is stored

Chapter 7 Chapter Table of Contents OVERVIEW...193 GETTING STARTED...194 TheThreeStagesofARIMAModeling...194 IdentificationStage...194 Estimation and Diagnostic Checking Stage...... 200 Forecasting Stage...205

### Time 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

### Econometrics Simple Linear Regression

Econometrics Simple Linear Regression Burcu Eke UC3M Linear equations with one variable Recall what a linear equation is: y = b 0 + b 1 x is a linear equation with one variable, or equivalently, a straight

### TIME SERIES ANALYSIS

TIME SERIES ANALYSIS L.M. BHAR AND V.K.SHARMA Indian Agricultural Statistics Research Institute Library Avenue, New Delhi-0 02 lmb@iasri.res.in. Introduction Time series (TS) data refers to observations

### TEMPORAL CAUSAL RELATIONSHIP BETWEEN STOCK MARKET CAPITALIZATION, TRADE OPENNESS AND REAL GDP: EVIDENCE FROM THAILAND

I J A B E R, Vol. 13, No. 4, (2015): 1525-1534 TEMPORAL CAUSAL RELATIONSHIP BETWEEN STOCK MARKET CAPITALIZATION, TRADE OPENNESS AND REAL GDP: EVIDENCE FROM THAILAND Komain Jiranyakul * Abstract: This study

### The 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

### Econometric Modelling for Revenue Projections

Econometric Modelling for Revenue Projections Annex E 1. An econometric modelling exercise has been undertaken to calibrate the quantitative relationship between the five major items of government revenue

### 5. Multiple regression

5. Multiple regression QBUS6840 Predictive Analytics https://www.otexts.org/fpp/5 QBUS6840 Predictive Analytics 5. Multiple regression 2/39 Outline Introduction to multiple linear regression Some useful

### Chapter 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

### Univariate 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

### Testing 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

### Internet 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

### Testing The Quantity Theory of Money in Greece: A Note

ERC Working Paper in Economic 03/10 November 2003 Testing The Quantity Theory of Money in Greece: A Note Erdal Özmen Department of Economics Middle East Technical University Ankara 06531, Turkey ozmen@metu.edu.tr

### Overview 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

### Contemporaneous Spill-over among Equity, Gold, and Exchange Rate Implied Volatility Indices

Contemporaneous Spill-over among Equity, Gold, and Exchange Rate Implied Volatility Indices Ihsan Ullah Badshah, Bart Frijns*, Alireza Tourani-Rad Department of Finance, Faculty of Business and Law, Auckland

### What 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

### Business Cycles and Natural Gas Prices

Department of Economics Discussion Paper 2004-19 Business Cycles and Natural Gas Prices Apostolos Serletis Department of Economics University of Calgary Canada and Asghar Shahmoradi Department of Economics

### Note 2 to Computer class: Standard mis-specification tests

Note 2 to Computer class: Standard mis-specification tests Ragnar Nymoen September 2, 2013 1 Why mis-specification testing of econometric models? As econometricians we must relate to the fact that the

### Business cycles and natural gas prices

Business cycles and natural gas prices Apostolos Serletis and Asghar Shahmoradi Abstract This paper investigates the basic stylised facts of natural gas price movements using data for the period that natural

### Integrated Resource Plan

Integrated Resource Plan March 19, 2004 PREPARED FOR KAUA I ISLAND UTILITY COOPERATIVE LCG Consulting 4962 El Camino Real, Suite 112 Los Altos, CA 94022 650-962-9670 1 IRP 1 ELECTRIC LOAD FORECASTING 1.1

### Time Series Analysis

JUNE 2012 Time Series Analysis CONTENT A time series is a chronological sequence of observations on a particular variable. Usually the observations are taken at regular intervals (days, months, years),

### Chapter 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

### Trend and Seasonal Components

Chapter 2 Trend and Seasonal Components If the plot of a TS reveals an increase of the seasonal and noise fluctuations with the level of the process then some transformation may be necessary before doing

### 17. SIMPLE LINEAR REGRESSION II

17. SIMPLE LINEAR REGRESSION II The Model In linear regression analysis, we assume that the relationship between X and Y is linear. This does not mean, however, that Y can be perfectly predicted from X.

### Is 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

### Forecasting in supply chains

1 Forecasting in supply chains Role of demand forecasting Effective transportation system or supply chain design is predicated on the availability of accurate inputs to the modeling process. One of the

### Government bond market linkages: evidence from Europe

Applied Financial Economics, 2005, 15, 599 610 Government bond market linkages: evidence from Europe Jian Yang Department of Accounting, Finance & MIS, Prairie View A&M University, Prairie View, TX 77446,

### Multivariate time series analysis is used when one wants to model and explain the interactions and comovements among a group of time series variables:

Multivariate Time Series Consider n time series variables {y 1t },...,{y nt }. A multivariate time series is the (n 1) vector time series {Y t } where the i th row of {Y t } is {y it }.Thatis,for any time

### Can we rely upon fiscal policy estimates in countries with a tax evasion of 15% of GDP?

Can we rely upon fiscal policy estimates in countries with a tax evasion of 15% of GDP? Raffaella Basile, Ministry of Economy and Finance, Dept. of Treasury Bruno Chiarini, University of Naples Parthenope,

### INDIRECT INFERENCE (prepared for: The New Palgrave Dictionary of Economics, Second Edition)

INDIRECT INFERENCE (prepared for: The New Palgrave Dictionary of Economics, Second Edition) Abstract Indirect inference is a simulation-based method for estimating the parameters of economic models. Its

### Working Paper. An Exploratory Time Series Analysis of Apprehensions and Linewatch Hours on the Southwest Border. Executive Summary

Working Paper AUGUST 2009 An Exploratory Time Series Analysis of Apprehensions and Linewatch Hours on the Southwest Border DEREKH CORNWELL, Ph.D.* Executive Summary Whether heightened border enforcement

### FORECASTING DEPOSIT GROWTH: Forecasting BIF and SAIF Assessable and Insured Deposits

Technical Paper Series Congressional Budget Office Washington, DC FORECASTING DEPOSIT GROWTH: Forecasting BIF and SAIF Assessable and Insured Deposits Albert D. Metz Microeconomic and Financial Studies

### Rob J Hyndman. Forecasting using. 11. Dynamic regression OTexts.com/fpp/9/1/ Forecasting using R 1

Rob J Hyndman Forecasting using 11. Dynamic regression OTexts.com/fpp/9/1/ Forecasting using R 1 Outline 1 Regression with ARIMA errors 2 Example: Japanese cars 3 Using Fourier terms for seasonality 4

### MULTIPLE 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

### Clustering in the Linear Model

Short Guides to Microeconometrics Fall 2014 Kurt Schmidheiny Universität Basel Clustering in the Linear Model 2 1 Introduction Clustering in the Linear Model This handout extends the handout on The Multiple

### Introduction 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

### Time 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.:

### Relationship among crude oil prices, share prices and exchange rates

Relationship among crude oil prices, share prices and exchange rates Do higher share prices and weaker dollar lead to higher crude oil prices? Akira YANAGISAWA Leader Energy Demand, Supply and Forecast

### Machine Learning in Statistical Arbitrage

Machine Learning in Statistical Arbitrage Xing Fu, Avinash Patra December 11, 2009 Abstract We apply machine learning methods to obtain an index arbitrage strategy. In particular, we employ linear regression

### Introduction 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

### Department of Economics

Department of Economics On Testing for Diagonality of Large Dimensional Covariance Matrices George Kapetanios Working Paper No. 526 October 2004 ISSN 1473-0278 On Testing for Diagonality of Large Dimensional

### 3. Regression & Exponential Smoothing

3. Regression & Exponential Smoothing 3.1 Forecasting a Single Time Series Two main approaches are traditionally used to model a single time series z 1, z 2,..., z n 1. Models the observation z t as a

### The Orthogonal Response of Stock Returns to Dividend Yield and Price-to-Earnings Innovations

The Orthogonal Response of Stock Returns to Dividend Yield and Price-to-Earnings Innovations Vichet Sum School of Business and Technology, University of Maryland, Eastern Shore Kiah Hall, Suite 2117-A

### The Dynamic Effects of Personal and Corporate Income Tax Changes in the United States

The Dynamic Effects of Personal and Corporate Income Tax Changes in the United States Karel Mertens and Morten O. Ravn March, Abstract This paper estimates the dynamic effects of changes in taxes in the

### Explaining Cointegration Analysis: Part II

Explaining Cointegration Analysis: Part II David F. Hendry and Katarina Juselius Nuffield College, Oxford, OX1 1NF. Department of Economics, University of Copenhagen, Denmark Abstract We describe the concept

### Chapter 2. Dynamic panel data models

Chapter 2. Dynamic panel data models Master of Science in Economics - University of Geneva Christophe Hurlin, Université d Orléans Université d Orléans April 2010 Introduction De nition We now consider

### Threshold Autoregressive Models in Finance: A Comparative Approach

University of Wollongong Research Online Applied Statistics Education and Research Collaboration (ASEARC) - Conference Papers Faculty of Informatics 2011 Threshold Autoregressive Models in Finance: A Comparative

### 1 Example of Time Series Analysis by SSA 1

1 Example of Time Series Analysis by SSA 1 Let us illustrate the 'Caterpillar'-SSA technique [1] by the example of time series analysis. Consider the time series FORT (monthly volumes of fortied wine sales

### The Use of Event Studies in Finance and Economics. Fall 2001. Gerald P. Dwyer, Jr.

The Use of Event Studies in Finance and Economics University of Rome at Tor Vergata Fall 2001 Gerald P. Dwyer, Jr. Any views are the author s and not necessarily those of the Federal Reserve Bank of Atlanta

### Simple 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

### Multivariate Normal Distribution

Multivariate Normal Distribution Lecture 4 July 21, 2011 Advanced Multivariate Statistical Methods ICPSR Summer Session #2 Lecture #4-7/21/2011 Slide 1 of 41 Last Time Matrices and vectors Eigenvalues

### The Effect of Infrastructure on Long Run Economic Growth

November, 2004 The Effect of Infrastructure on Long Run Economic Growth David Canning Harvard University and Peter Pedroni * Williams College --------------------------------------------------------------------------------------------------------------------

### Implied volatility transmissions between Thai and selected advanced stock markets

MPRA Munich Personal RePEc Archive Implied volatility transmissions between Thai and selected advanced stock markets Supachok Thakolsri and Yuthana Sethapramote and Komain Jiranyakul Public Enterprise

### THE EFFECTS OF BANKING CREDIT ON THE HOUSE PRICE

THE EFFECTS OF BANKING CREDIT ON THE HOUSE PRICE * Adibeh Savari 1, Yaser Borvayeh 2 1 MA Student, Department of Economics, Science and Research Branch, Islamic Azad University, Khuzestan, Iran 2 MA Student,

### Dynamics of Real Investment and Stock Prices in Listed Companies of Tehran Stock Exchange

Dynamics of Real Investment and Stock Prices in Listed Companies of Tehran Stock Exchange Farzad Karimi Assistant Professor Department of Management Mobarakeh Branch, Islamic Azad University, Mobarakeh,

### Module 5: Multiple Regression Analysis

Using Statistical Data Using to Make Statistical Decisions: Data Multiple to Make Regression Decisions Analysis Page 1 Module 5: Multiple Regression Analysis Tom Ilvento, University of Delaware, College

### Corporate 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

### Sales forecasting # 1

Sales forecasting # 1 Arthur Charpentier arthur.charpentier@univ-rennes1.fr 1 Agenda Qualitative and quantitative methods, a very general introduction Series decomposition Short versus long term forecasting

### Impact of foreign portfolio investments on market comovements: Evidence from the emerging Indian stock market

Impact of foreign portfolio investments on market comovements: Evidence from the emerging Indian stock market Sunil Poshakwale and Chandra Thapa Cranfield School of Management, Cranfield University, Cranfield,

### VOLATILITY SPILLOVERS AMONG ALTERNATIVE ENERGY, OIL AND TECHNOLOGY GLOBAL MARKETS. Elena Renedo Sánchez

VOLATILITY SPILLOVERS AMONG ALTERNATIVE ENERGY, OIL AND TECHNOLOGY GLOBAL MARKETS Elena Renedo Sánchez Trabajo de investigación 015/015 Master en Banca y Finanzas Cuantitativas Tutora: Dra. Pilar Soriano

### E 4101/5101 Lecture 8: Exogeneity

E 4101/5101 Lecture 8: Exogeneity Ragnar Nymoen 17 March 2011 Introduction I Main references: Davidson and MacKinnon, Ch 8.1-8,7, since tests of (weak) exogeneity build on the theory of IV-estimation Ch

### Forecasting the US Dollar / Euro Exchange rate Using ARMA Models

Forecasting the US Dollar / Euro Exchange rate Using ARMA Models LIUWEI (9906360) - 1 - ABSTRACT...3 1. INTRODUCTION...4 2. DATA ANALYSIS...5 2.1 Stationary estimation...5 2.2 Dickey-Fuller Test...6 3.

### Data Mining and Data Warehousing. Henryk Maciejewski. Data Mining Predictive modelling: regression

Data Mining and Data Warehousing Henryk Maciejewski Data Mining Predictive modelling: regression Algorithms for Predictive Modelling Contents Regression Classification Auxiliary topics: Estimation of prediction

### Predictable Dividends and Returns

Predictable Dividends and Returns Job Market Paper Ruy Ribeiro 1 Graduate School of Business University of Chicago November, 2002 1 Ph.D. Candidate in Finance. Email: ruy.ribeiro@gsb.uchicago.edu. I am

### Centre 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

### Monetary Policy and Long-term Interest Rates

Monetary Policy and Long-term Interest Rates Shu Wu The University of Kansas First Draft: July 2004 This version: March 2005 Abstract This paper documents some new empirical results about the monetary

### Singular Spectrum Analysis with Rssa

Singular Spectrum Analysis with Rssa Maurizio Sanarico Chief Data Scientist SDG Consulting Milano R June 4, 2014 Motivation Discover structural components in complex time series Working hypothesis: a signal

### Masters in Financial Economics (MFE)

Masters in Financial Economics (MFE) Admission Requirements Candidates must submit the following to the Office of Admissions and Registration: 1. Official Transcripts of previous academic record 2. Two

### Elucidating the Relationship among Volatility Index, US Dollar Index and Oil Price

23-24 July 25, Sheraton LaGuardia East Hotel, New York, USA, ISBN: 978--92269-79-5 Elucidating the Relationship among Volatility Index, US Dollar Index and Oil Price John Wei-Shan Hu* and Hsin-Yi Chang**

### CHAPTER 11: ARBITRAGE PRICING THEORY

CHAPTER 11: ARBITRAGE PRICING THEORY 1. The revised estimate of the expected rate of return on the stock would be the old estimate plus the sum of the products of the unexpected change in each factor times

### The Long-Run Relation Between The Personal Savings Rate And Consumer Sentiment

The Long-Run Relation Between The Personal Savings Rate And Consumer Sentiment Bradley T. Ewing 1 and James E. Payne 2 This study examined the long run relationship between the personal savings rate and

### Trends and Breaks in Cointegrated VAR Models

Trends and Breaks in Cointegrated VAR Models Håvard Hungnes Thesis for the Dr. Polit. degree Department of Economics, University of Oslo Defended March 17, 2006 Research Fellow in the Research Department

### NCSS Statistical Software Principal Components Regression. In ordinary least squares, the regression coefficients are estimated using the formula ( )

Chapter 340 Principal Components Regression Introduction is a technique for analyzing multiple regression data that suffer from multicollinearity. When multicollinearity occurs, least squares estimates

### MATHEMATICAL METHODS OF STATISTICS

MATHEMATICAL METHODS OF STATISTICS By HARALD CRAMER TROFESSOK IN THE UNIVERSITY OF STOCKHOLM Princeton PRINCETON UNIVERSITY PRESS 1946 TABLE OF CONTENTS. First Part. MATHEMATICAL INTRODUCTION. CHAPTERS

### Demand Forecasting When a product is produced for a market, the demand occurs in the future. The production planning cannot be accomplished unless

Demand Forecasting When a product is produced for a market, the demand occurs in the future. The production planning cannot be accomplished unless the volume of the demand known. The success of the business

### Co-movements of NAFTA trade, FDI and stock markets

Co-movements of NAFTA trade, FDI and stock markets Paweł Folfas, Ph. D. Warsaw School of Economics Abstract The paper scrutinizes the causal relationship between performance of American, Canadian and Mexican

### IS THERE A LONG-RUN RELATIONSHIP

7. IS THERE A LONG-RUN RELATIONSHIP BETWEEN TAXATION AND GROWTH: THE CASE OF TURKEY Salih Turan KATIRCIOGLU Abstract This paper empirically investigates long-run equilibrium relationship between economic

### Kiwi drivers the New Zealand dollar experience AN 2012/ 02

Kiwi drivers the New Zealand dollar experience AN 2012/ 02 Chris McDonald May 2012 Reserve Bank of New Zealand Analytical Note series ISSN 2230-5505 Reserve Bank of New Zealand PO Box 2498 Wellington NEW

### Time Series Analysis: Basic Forecasting.

Time Series Analysis: Basic Forecasting. As published in Benchmarks RSS Matters, April 2015 http://web3.unt.edu/benchmarks/issues/2015/04/rss-matters Jon Starkweather, PhD 1 Jon Starkweather, PhD jonathan.starkweather@unt.edu