# Part 1: White Noise and Moving Average Models

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Chaper 3: Forecasing From Time Series Models Par 1: Whie Noise and Moving Average Models Saionariy In his chaper, we sudy models for saionary ime series. A ime series is saionary if is underlying saisical srucure does no evolve wih ime. A saionary series is unlikely o exhibi long-erm rends. To see why, we need a beer definiion of rend. Trend is a endency of he series o increase (or decrease) no necessarily for he realizaion ha acually occurred, bu insead for he "ypical" realizaion. Consider he underlying (populaion) mean of he series, E [x ]. This represens he average value we would ge for he series a ime if we zaion, bu he key aspec of saisical hinking is o admi ha he daa series we acually go is jus a could urn back he hands of ime and look a many realizaions of he series. (We only have one realisample realizaion from he many series we migh have goen bu didn.) Since E [x ] is deermined by he underlying saisical srucure of he series, saionariy implies ha E [x ] mus no depend on ime. Bu his means ha he series has no underlying rend. applied. Since saionary ime series are unlikely o exhibi long-erm rends, any apparen rend in he daa should firs be removed (e.g. by differencing) if he models discussed here are o be usefully Auocorrelaion A useful measure of linear associaion beween wo random variables X and Y (assumed here o have mean zero) is heir correlaion : Corr (X, Y ) = Covariance (X, Y ) = E [XY ]. Variance (X ) Variance (Y ) E [X ] E [Y ] I is always rue ha 1 Corr (X, Y ) 1. The larger he absolue value of Corr (X, Y ), he sronger he linear associaion beween X and Y. The concep of correlaion plays a key role in ime series analysis. We hink of x

2 -- ( = 1,,3,... ) as a sequence of random variables, any pair of which may be correlaed. The saionariy assumpion implies ha Corr (x, x since i is a correlaion beween he ime series and is own pas. j ) depends only on he ime separaion j, and no on he ime locaion. Noe ha Corr (x, x ) = 1. We ofen refer o Corr (x, x j )asanauocorrelaion As long as Corr (x, x ) 0 for some j > 0, we can obain a linear forecas of x from j x 1, x,.... To give a simple example of his, suppose ha Corr (x, x 1) =ρ 1, and ha E [x ] = E [x ] = 0. Since ρ is a correlaion, we mus have 1 ρ 1. The larger ρ, he sronger he linear associaion beween x and x 1. The bes linear predicor of x based on x 1 is ρ1x 1. Thus, if ρ >0(posiive auocorrelaion ), he forecas for x increases as x 1 1 increases. Examples include he daily NASDAQ index, daily emperaures, monhly ineres raes. If ρ <0(negaive auocorrelaion ) 1, he forecas for x decreases as x increases. Examples include hours of sleep per nigh, he daily 1 caloric inake of an individual, and oupu from a producion process which is being coninuously adjused o achieve a desired arge oupu. If ρ =0, hen x and x are said o be uncorrelaed,and 1 1 he bes linear forecas of x is zero (or, in general, E [x ]). Examples include he daily reurns on Gen- eral Moors sock, and he difference beween 3.5 and he number rolled in independen osses of a fair die. Noe ha, if x and x are uncorrelaed, knowledge of x does no help us o linearly forecas x. 1 1 Whie Noise A saionary ime series ε is said o be whie noise if Corr (ε, ε ) = 0 for all s. s Thus, ε is a sequence of uncorrelaed random variables wih consan variance and consan mean. We will assume ha his consan mean value is zero. Plos of whie noise series exhibi a very erraic, jumpy, unpredicable behavior. Since he ε are uncorrelaed, previous values do no help us o forecas fuure values. The resuls of successive spins of a roulee wheel provide an example of a whie noise series. Whie noise series hemselves are quie unineresing from a forecasing sandpoin (hey are no linearly forecasable), bu hey form he building blocks for more general models. In economic ime series, he whie noise series is ofen hough of as represening innovaions, or

3 -3- shocks. Tha is, ε represens hose aspecs of he ime series of ineres which could no have been prediced in advance. Moving Averages A simple moving average is a series x generaed from a whie noise series ε by he rule x =ε +βε 1. Noe ha, unless β=0, x will have a nonrivial correlaion srucure. In fac, we have Corr (x, x 1 var ε ) =β var x, (1) Corr (x, x j ) = 0 (j > 1). () Equaion (1) implies ha if β is posiive, hen adjacen erms of x will be posiively correlaed, so ha an above average (i.e., posiive) x will end o be followed by a furher above average value. For a whie noise series, β=0, an above average value is equally likely o be followed by an above average or a below average value. If β is negaive, hen an above average erm is likely o be followed by a below average erm. We see ha if β is posiive (zero or negaive), hen x will be smooher (as smooh or rougher) han he whie noise series ε. Equaion () implies ha here is no correlaion beween he presen value of x and all previous values apar from he mos recen. Thus, he simple moving average is said o have a shor memory. To prove equaion (1), noe ha E [x x ] = E [(ε + βε )(ε + βε )] = E [ε ε ] +βe [ε ] +β E [ε ε ] +βe [ε ε 1 ] 1 =βe [ε ] s j ince E [ε ε ] = 0 for all j > 0. Since ε is saionary wih zero mean, so is x, and we have 1 E [ε ] = var ε = var ε, var x = var x 1 1.

4 -4- Thus, Corr [x, x ] = 1 E [x x 1] = var x var x 1 βe [ε 1] = var x var x 1 β var ε. var x The proof of () is similar. of x Suppose we wan o forecas xn +1 from x 1,...,xn for he simple moving average x =ε +βε 1 wih β known. Since x =ε +βε and since he bes forecas of ε is zero, he opimal forecas n +1 is n +1 n +1 n n+1 f n,1 =βεn. To obain a useful forecas, however, we mus express ε in erms of x,...,x. Since f n 1,1 n 1 n n n 1 =βε and since x =ε +βε, we have n 1 n ε = x βε = x f. n n n 1 n n 1,1 If we (somewha arbirarily) se f = 0, hen we can compue (approximaions o) ε, ε,...,ε b recursively applying he formula 0,1 1 n y ε = x f, = 1,,...,n. 1,1 We can hen evaluae he forecas f =βε, n,1 n a n 1,1 1,1 s well as f,...,f. A measure of forecasabiliy of a series is R = 1 Variance of Forecas Error. Variance of x This analogous o he version of R used in regression analysis, R = 1 SS /SS. Jus as in regres Resid Toal - sion, he forecasing version of R is guaraneed o be beween 0 and 1. If R is close o zero, hen he variance of he forecas error is almos as large as he variance of he series x iself. Thus, he bes linear forecas is no performing much beer han he rivial forecas (zero), so he series is no very forecasable when R is close o zero. On he oher hand, if R is close o 1, hen he forecas error

5 -5- from he bes forecas is much less han he forecas error from he rivial forecas, so he series is very forecasable in his case. For he simple moving average, we have var x = var (ε +βε ) 1 = var ε + β var ε + βcov (ε, ε 1 1) = (1 +β ) var ε. The forecas error is e = x f = [ε + βε ] βε =ε. n,1 n +1 n,1 n +1 n n n+1 Thus, R reduces o R = 1 var ε = 1 1 = β. (1 +β )var ε 1 +β 1 +β Forecass for more han one sep ahead are all zero, since for h > 1 we have x =ε +βε, n +h n+h n+h 1 boh erms of which are "unforecasable". In his case, R = 0. The simple moving average is also called a f irs order moving average, denoed by MA(1), because i conains jus one parameer, β. A generalizaion of he MA(1) model is he q h order moving average MA(q), given by x =ε +βε +βε β ε. 1 1 q q I is easily shown ha for he MA(q) series, and hence he forecas f n, h Corr (x, x ) = 0 j > q j is zero for h > q. Since he opimal one-sep forecas is x =ε +βε +βε β ε, n +1 n +1 1 n n 1 q n q +1 f n,1 =βε 1 n +βε n βqεn q + 1

6 -6- and he one-sep forecas error is To acually f orm he opimum forecas, sar wih f e = x f =ε. n,1 n +1 n,1 n +1 0, 1 = 0 and hen form εˆ = x f, = 1,,...,n. 1,1 In general, o form he forecas f for j q, wrie down he expression for x, replace every unk n, j n+j - nowable fuure value of ε by 0 and all he remaining values by he εˆ given above. How migh a moving average model arise in he real world? For example, suppose y is he daily change in price of some produc, and ε +1 is he effec on omorrow s price of unexpeced news. The full impac of his news may no be compleely absorbed by he marke. Suppose his akes wo days o happen. Then we would have y =ε +b ε, where ε is he news which canno be prediced from ime +1, and b ε +1 is he reassessmen of he earlier piece of news. This would lead o an MA (1) model for y. Anoher way ha an MA (1) model can arise is hrough overdif f erencing. Suppose, for example, ha our original ime series is whie noise, x =ε. This is saionary, and should no be differenced. Differencing a series which is already saionary is called overdif f erencing and should be avoided if possible, bu i ofen occurs by acciden. If we (over) difference our whie noise series, we ge y =ε ε, which is an MA (1) series, wih negaive auocorrelaion. Alhough he original series was 1 no linearly predicable, he differenced series will be!

### Usefulness of the Forward Curve in Forecasting Oil Prices

Usefulness of he Forward Curve in Forecasing Oil Prices Akira Yanagisawa Leader Energy Demand, Supply and Forecas Analysis Group The Energy Daa and Modelling Cener Summary When people analyse oil prices,

### Chapter 8 Student Lecture Notes 8-1

Chaper Suden Lecure Noes - Chaper Goals QM: Business Saisics Chaper Analyzing and Forecasing -Series Daa Afer compleing his chaper, you should be able o: Idenify he componens presen in a ime series Develop

### Lecture 18. Serial correlation: testing and estimation. Testing for serial correlation

Lecure 8. Serial correlaion: esing and esimaion Tesing for serial correlaion In lecure 6 we used graphical mehods o look for serial/auocorrelaion in he random error erm u. Because we canno observe he u

### Lecture 12 Assumption Violation: Autocorrelation

Major Topics: Definiion Lecure 1 Assumpion Violaion: Auocorrelaion Daa Relaionship Represenaion Problem Deecion Remedy Page 1.1 Our Usual Roadmap Parial View Expansion of Esimae and Tes Model Sep Analyze

### YTM is positively related to default risk. YTM is positively related to liquidity risk. YTM is negatively related to special tax treatment.

. Two quesions for oday. A. Why do bonds wih he same ime o mauriy have differen YTM s? B. Why do bonds wih differen imes o mauriy have differen YTM s? 2. To answer he firs quesion les look a he risk srucure

### INVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS

INVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS Ilona Tregub, Olga Filina, Irina Kondakova Financial Universiy under he Governmen of he Russian Federaion 1. Phillips curve In economics,

### An empirical analysis about forecasting Tmall air-conditioning sales using time series model Yan Xia

An empirical analysis abou forecasing Tmall air-condiioning sales using ime series model Yan Xia Deparmen of Mahemaics, Ocean Universiy of China, China Absrac Time series model is a hospo in he research

### Issues Using OLS with Time Series Data. Time series data NOT randomly sampled in same way as cross sectional each obs not i.i.d

These noes largely concern auocorrelaion Issues Using OLS wih Time Series Daa Recall main poins from Chaper 10: Time series daa NOT randomly sampled in same way as cross secional each obs no i.i.d Why?

### Vector Autoregressions (VARs): Operational Perspectives

Vecor Auoregressions (VARs): Operaional Perspecives Primary Source: Sock, James H., and Mark W. Wason, Vecor Auoregressions, Journal of Economic Perspecives, Vol. 15 No. 4 (Fall 2001), 101-115. Macroeconomericians

### Graphing the Von Bertalanffy Growth Equation

file: d:\b173-2013\von_beralanffy.wpd dae: Sepember 23, 2013 Inroducion Graphing he Von Beralanffy Growh Equaion Previously, we calculaed regressions of TL on SL for fish size daa and ploed he daa and

### Revisions to Nonfarm Payroll Employment: 1964 to 2011

Revisions o Nonfarm Payroll Employmen: 1964 o 2011 Tom Sark December 2011 Summary Over recen monhs, he Bureau of Labor Saisics (BLS) has revised upward is iniial esimaes of he monhly change in nonfarm

### Forecasting Malaysian Gold Using. GARCH Model

Applied Mahemaical Sciences, Vol. 7, 2013, no. 58, 2879-2884 HIKARI Ld, www.m-hikari.com Forecasing Malaysian Gold Using GARCH Model Pung Yean Ping 1, Nor Hamizah Miswan 2 and Maizah Hura Ahmad 3 Deparmen

### 4. International Parity Conditions

4. Inernaional ariy ondiions 4.1 urchasing ower ariy he urchasing ower ariy ( heory is one of he early heories of exchange rae deerminaion. his heory is based on he concep ha he demand for a counry's currency

### 11/6/2013. Chapter 14: Dynamic AD-AS. Introduction. Introduction. Keeping track of time. The model s elements

Inroducion Chaper 14: Dynamic D-S dynamic model of aggregae and aggregae supply gives us more insigh ino how he economy works in he shor run. I is a simplified version of a DSGE model, used in cuing-edge

### ... t t t k kt t. We assume that there are observations covering n periods of time. Autocorrelation (also called

Supplemen 13B: Durbin-Wason Tes for Auocorrelaion Modeling Auocorrelaion Because auocorrelaion is primarily a phenomenon of ime series daa, i is convenien o represen he linear regression model using as

### CHARGE AND DISCHARGE OF A CAPACITOR

REFERENCES RC Circuis: Elecrical Insrumens: Mos Inroducory Physics exs (e.g. A. Halliday and Resnick, Physics ; M. Sernheim and J. Kane, General Physics.) This Laboraory Manual: Commonly Used Insrumens:

### Cointegration: The Engle and Granger approach

Coinegraion: The Engle and Granger approach Inroducion Generally one would find mos of he economic variables o be non-saionary I(1) variables. Hence, any equilibrium heories ha involve hese variables require

### cooking trajectory boiling water B (t) microwave 0 2 4 6 8 101214161820 time t (mins)

Alligaor egg wih calculus We have a large alligaor egg jus ou of he fridge (1 ) which we need o hea o 9. Now here are wo accepable mehods for heaing alligaor eggs, one is o immerse hem in boiling waer

### A Brief Introduction to the Consumption Based Asset Pricing Model (CCAPM)

A Brief Inroducion o he Consumpion Based Asse Pricing Model (CCAPM We have seen ha CAPM idenifies he risk of any securiy as he covariance beween he securiy's rae of reurn and he rae of reurn on he marke

### DIFFERENCING AND UNIT ROOT TESTS

DIFFERENCING AND UNIT ROOT TESTS In he Box-Jenkins approach o analyzing ime series, a key quesion is wheher o difference he daa, i.e., o replace he raw daa {x } by he differenced series {x x }. Experience

### The naive method discussed in Lecture 1 uses the most recent observations to forecast future values. That is, Y ˆ t + 1

Business Condiions & Forecasing Exponenial Smoohing LECTURE 2 MOVING AVERAGES AND EXPONENTIAL SMOOTHING OVERVIEW This lecure inroduces ime-series smoohing forecasing mehods. Various models are discussed,

### Chapter 8: Regression with Lagged Explanatory Variables

Chaper 8: Regression wih Lagged Explanaory Variables Time series daa: Y for =1,..,T End goal: Regression model relaing a dependen variable o explanaory variables. Wih ime series new issues arise: 1. One

### 4. The Poisson Distribution

Virual Laboraories > 13. The Poisson Process > 1 2 3 4 5 6 7 4. The Poisson Disribuion The Probabiliy Densiy Funcion We have shown ha he k h arrival ime in he Poisson process has he gamma probabiliy densiy

### Chapter 7. Response of First-Order RL and RC Circuits

Chaper 7. esponse of Firs-Order L and C Circuis 7.1. The Naural esponse of an L Circui 7.2. The Naural esponse of an C Circui 7.3. The ep esponse of L and C Circuis 7.4. A General oluion for ep and Naural

### Newton's second law in action

Newon's second law in acion In many cases, he naure of he force acing on a body is known I migh depend on ime, posiion, velociy, or some combinaion of hese, bu is dependence is known from experimen In

### 4.8 Exponential Growth and Decay; Newton s Law; Logistic Growth and Decay

324 CHAPTER 4 Exponenial and Logarihmic Funcions 4.8 Exponenial Growh and Decay; Newon s Law; Logisic Growh and Decay OBJECTIVES 1 Find Equaions of Populaions Tha Obey he Law of Uninhibied Growh 2 Find

### PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART TWO

Profi Tes Modelling in Life Assurance Using Spreadshees, par wo PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART TWO Erik Alm Peer Millingon Profi Tes Modelling in Life Assurance Using Spreadshees,

### Renewal processes and Poisson process

CHAPTER 3 Renewal processes and Poisson process 31 Definiion of renewal processes and limi heorems Le ξ 1, ξ 2, be independen and idenically disribued random variables wih P[ξ k > 0] = 1 Define heir parial

### Economics 140A Hypothesis Testing in Regression Models

Economics 140A Hypohesis Tesing in Regression Models While i is algebraically simple o work wih a populaion model wih a single varying regressor, mos populaion models have muliple varying regressors 1

### Impact of Debt on Primary Deficit and GSDP Gap in Odisha: Empirical Evidences

S.R. No. 002 10/2015/CEFT Impac of Deb on Primary Defici and GSDP Gap in Odisha: Empirical Evidences 1. Inroducion The excessive pressure of public expendiure over is revenue receip is financed hrough

### MTH6121 Introduction to Mathematical Finance Lesson 5

26 MTH6121 Inroducion o Mahemaical Finance Lesson 5 Conens 2.3 Brownian moion wih drif........................... 27 2.4 Geomeric Brownian moion........................... 28 2.5 Convergence of random

### State Machines: Brief Introduction to Sequencers Prof. Andrew J. Mason, Michigan State University

Inroducion ae Machines: Brief Inroducion o equencers Prof. Andrew J. Mason, Michigan ae Universiy A sae machine models behavior defined by a finie number of saes (unique configuraions), ransiions beween

### DOES TRADING VOLUME INFLUENCE GARCH EFFECTS? SOME EVIDENCE FROM THE GREEK MARKET WITH SPECIAL REFERENCE TO BANKING SECTOR

Invesmen Managemen and Financial Innovaions, Volume 4, Issue 3, 7 33 DOES TRADING VOLUME INFLUENCE GARCH EFFECTS? SOME EVIDENCE FROM THE GREEK MARKET WITH SPECIAL REFERENCE TO BANKING SECTOR Ahanasios

### Fair games, and the Martingale (or "Random walk") model of stock prices

Economics 236 Spring 2000 Professor Craine Problem Se 2: Fair games, and he Maringale (or "Random walk") model of sock prices Sephen F LeRoy, 989. Efficien Capial Markes and Maringales, J of Economic Lieraure,27,

### Duration and Convexity ( ) 20 = Bond B has a maturity of 5 years and also has a required rate of return of 10%. Its price is \$613.

Graduae School of Business Adminisraion Universiy of Virginia UVA-F-38 Duraion and Convexiy he price of a bond is a funcion of he promised paymens and he marke required rae of reurn. Since he promised

### Chapter 2 Problems. s = d t up. = 40km / hr d t down. 60km / hr. d t total. + t down. = t up. = 40km / hr + d. 60km / hr + 40km / hr

Chaper 2 Problems 2.2 A car ravels up a hill a a consan speed of 40km/h and reurns down he hill a a consan speed of 60 km/h. Calculae he average speed for he rip. This problem is a bi more suble han i

### Chapter 4. Properties of the Least Squares Estimators. Assumptions of the Simple Linear Regression Model. SR3. var(e t ) = σ 2 = var(y t )

Chaper 4 Properies of he Leas Squares Esimaors Assumpions of he Simple Linear Regression Model SR1. SR. y = β 1 + β x + e E(e ) = 0 E[y ] = β 1 + β x SR3. var(e ) = σ = var(y ) SR4. cov(e i, e j ) = cov(y

### Relative velocity in one dimension

Connexions module: m13618 1 Relaive velociy in one dimension Sunil Kumar Singh This work is produced by The Connexions Projec and licensed under he Creaive Commons Aribuion License Absrac All quaniies

### Mathematics in Pharmacokinetics What and Why (A second attempt to make it clearer)

Mahemaics in Pharmacokineics Wha and Why (A second aemp o make i clearer) We have used equaions for concenraion () as a funcion of ime (). We will coninue o use hese equaions since he plasma concenraions

### SPECIAL REPORT May 4, Shifting Drivers of Inflation Canada versus the U.S.

Paul Ferley Assisan Chief Economis 416-974-7231 paul.ferley@rbc.com Nahan Janzen Economis 416-974-0579 nahan.janzen@rbc.com SPECIAL REPORT May 4, 2010 Shifing Drivers of Inflaion Canada versus he U.S.

### Chapter 7: Estimating the Variance of an Estimate s Probability Distribution

Chaper 7: Esimaing he Variance of an Esimae s Probabiliy Disribuion Chaper 7 Ouline Review o Clin s Assignmen o General Properies of he Ordinary Leas Squares (OLS) Esimaion Procedure o Imporance of he

### Fourier Series Solution of the Heat Equation

Fourier Series Soluion of he Hea Equaion Physical Applicaion; he Hea Equaion In he early nineeenh cenury Joseph Fourier, a French scienis and mahemaician who had accompanied Napoleon on his Egypian campaign,

### A Note on Using the Svensson procedure to estimate the risk free rate in corporate valuation

A Noe on Using he Svensson procedure o esimae he risk free rae in corporae valuaion By Sven Arnold, Alexander Lahmann and Bernhard Schwezler Ocober 2011 1. The risk free ineres rae in corporae valuaion

### Forecasting and Information Sharing in Supply Chains Under Quasi-ARMA Demand

Forecasing and Informaion Sharing in Supply Chains Under Quasi-ARMA Demand Avi Giloni, Clifford Hurvich, Sridhar Seshadri July 9, 2009 Absrac In his paper, we revisi he problem of demand propagaion in

### The Transport Equation

The Transpor Equaion Consider a fluid, flowing wih velociy, V, in a hin sraigh ube whose cross secion will be denoed by A. Suppose he fluid conains a conaminan whose concenraion a posiion a ime will be

### Digital Data Acquisition

ME231 Measuremens Laboraory Spring 1999 Digial Daa Acquisiion Edmundo Corona c The laer par of he 20h cenury winessed he birh of he compuer revoluion. The developmen of digial compuer echnology has had

### Measuring macroeconomic volatility Applications to export revenue data, 1970-2005

FONDATION POUR LES ETUDES ET RERS LE DEVELOPPEMENT INTERNATIONAL Measuring macroeconomic volailiy Applicaions o expor revenue daa, 1970-005 by Joël Cariolle Policy brief no. 47 March 01 The FERDI is a

### Understanding Sequential Circuit Timing

ENGIN112: Inroducion o Elecrical and Compuer Engineering Fall 2003 Prof. Russell Tessier Undersanding Sequenial Circui Timing Perhaps he wo mos disinguishing characerisics of a compuer are is processor

### Technical Description of S&P 500 Buy-Write Monthly Index Composition

Technical Descripion of S&P 500 Buy-Wrie Monhly Index Composiion The S&P 500 Buy-Wrie Monhly (BWM) index is a oal reurn index based on wriing he nearby a-he-money S&P 500 call opion agains he S&P 500 index

### Representing Periodic Functions by Fourier Series. (a n cos nt + b n sin nt) n=1

Represening Periodic Funcions by Fourier Series 3. Inroducion In his Secion we show how a periodic funcion can be expressed as a series of sines and cosines. We begin by obaining some sandard inegrals

### Basic Assumption: population dynamics of a group controlled by two functions of time

opulaion Models Basic Assumpion: populaion dynamics of a group conrolled by wo funcions of ime Birh Rae β(, ) = average number of birhs per group member, per uni ime Deah Rae δ(, ) = average number of

### Markov Models and Hidden Markov Models (HMMs)

Markov Models and Hidden Markov Models (HMMs (Following slides are modified from Prof. Claire Cardie s slides and Prof. Raymond Mooney s slides. Some of he graphs are aken from he exbook. Markov Model

### 4 Convolution. Recommended Problems. x2[n] 1 2[n]

4 Convoluion Recommended Problems P4.1 This problem is a simple example of he use of superposiion. Suppose ha a discree-ime linear sysem has oupus y[n] for he given inpus x[n] as shown in Figure P4.1-1.

### Signal Rectification

9/3/25 Signal Recificaion.doc / Signal Recificaion n imporan applicaion of juncion diodes is signal recificaion. here are wo ypes of signal recifiers, half-wae and fullwae. Le s firs consider he ideal

### Individual Health Insurance April 30, 2008 Pages 167-170

Individual Healh Insurance April 30, 2008 Pages 167-170 We have received feedback ha his secion of he e is confusing because some of he defined noaion is inconsisen wih comparable life insurance reserve

### DYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS

DYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS Hong Mao, Shanghai Second Polyechnic Universiy Krzyszof M. Osaszewski, Illinois Sae Universiy Youyu Zhang, Fudan Universiy ABSTRACT Liigaion, exper

### Forecasting, Ordering and Stock- Holding for Erratic Demand

ISF 2002 23 rd o 26 h June 2002 Forecasing, Ordering and Sock- Holding for Erraic Demand Andrew Eaves Lancaser Universiy / Andalus Soluions Limied Inroducion Erraic and slow-moving demand Demand classificaion

### Why Do Real and Nominal. Inventory-Sales Ratios Have Different Trends?

Why Do Real and Nominal Invenory-Sales Raios Have Differen Trends? By Valerie A. Ramey Professor of Economics Deparmen of Economics Universiy of California, San Diego and Research Associae Naional Bureau

### Chapter 2 Problems. 3600s = 25m / s d = s t = 25m / s 0.5s = 12.5m. Δx = x(4) x(0) =12m 0m =12m

Chaper 2 Problems 2.1 During a hard sneeze, your eyes migh shu for 0.5s. If you are driving a car a 90km/h during such a sneeze, how far does he car move during ha ime s = 90km 1000m h 1km 1h 3600s = 25m

### Section 7.1 Angles and Their Measure

Secion 7.1 Angles and Their Measure Greek Leers Commonly Used in Trigonomery Quadran II Quadran III Quadran I Quadran IV α = alpha β = bea θ = hea δ = dela ω = omega γ = gamma DEGREES The angle formed

### A Mathematical Description of MOSFET Behavior

10/19/004 A Mahemaical Descripion of MOSFET Behavior.doc 1/8 A Mahemaical Descripion of MOSFET Behavior Q: We ve learned an awful lo abou enhancemen MOSFETs, bu we sill have ye o esablished a mahemaical

### Graduate Macro Theory II: Notes on Neoclassical Growth Model

Graduae Macro Theory II: Noes on Neoclassical Growh Model Eric Sims Universiy of Nore Dame Spring 2011 1 Basic Neoclassical Growh Model The economy is populaed by a large number of infiniely lived agens.

### Economic Factors in Determining the Penetration Coefficient of Mobile Phone in Iran

Iranian Economic Review, Vol.5, No.26, Spring 200 Economic Facors in Deermining he Peneraion Coefficien of Mobile Phone in Iran Mansour Khalili Araghi Ghahreman Abdoli Absrac n his paper we have sudied

### Chapter 6. First Order PDEs. 6.1 Characteristics The Simplest Case. u(x,t) t=1 t=2. t=0. Suppose u(x, t) satisfies the PDE.

Chaper 6 Firs Order PDEs 6.1 Characerisics 6.1.1 The Simples Case Suppose u(, ) saisfies he PDE where b, c are consan. au + bu = 0 If a = 0, he PDE is rivial (i says ha u = 0 and so u = f(). If a = 0,

### CHAPTER 12: AUTOCORRELATION: WHAT HAPPENS IF THE ERROR TERMS ARE CORRELATED?

Basic Economerics, Gujarai and Porer CHAPTER 1: AUTOCORRELATION: WHAT HAPPENS IF THE ERROR TERMS ARE CORRELATED? 1.1 (a) False. The esimaors are unbiased bu hey are no efficien. (b) True. We are sill reaining

### Supply Chain Management Using Simulation Optimization By Miheer Kulkarni

Supply Chain Managemen Using Simulaion Opimizaion By Miheer Kulkarni This problem was inspired by he paper by Jung, Blau, Pekny, Reklaii and Eversdyk which deals wih supply chain managemen for he chemical

### The Greek financial crisis: growing imbalances and sovereign spreads. Heather D. Gibson, Stephan G. Hall and George S. Tavlas

The Greek financial crisis: growing imbalances and sovereign spreads Heaher D. Gibson, Sephan G. Hall and George S. Tavlas The enry The enry of Greece ino he Eurozone in 2001 produced a dividend in he

### The Real Business Cycle paradigm. The RBC model emphasizes supply (technology) disturbances as the main source of

Prof. Harris Dellas Advanced Macroeconomics Winer 2001/01 The Real Business Cycle paradigm The RBC model emphasizes supply (echnology) disurbances as he main source of macroeconomic flucuaions in a world

### Technical Appendix to Risk, Return, and Dividends

Technical Appendix o Risk, Reurn, and Dividends Andrew Ang Columbia Universiy and NBER Jun Liu UC San Diego This Version: 28 Augus, 2006 Columbia Business School, 3022 Broadway 805 Uris, New York NY 10027,

### Research Question Is the average body temperature of healthy adults 98.6 F? Introduction to Hypothesis Testing. Statistical Hypothesis

Inroducion o Hypohesis Tesing Research Quesion Is he average body emperaure of healhy aduls 98.6 F? HT - 1 HT - 2 Scienific Mehod 1. Sae research hypoheses or quesions. µ = 98.6? 2. Gaher daa or evidence

### Random Walk in 1-D. 3 possible paths x vs n. -5 For our random walk, we assume the probabilities p,q do not depend on time (n) - stationary

Random Walk in -D Random walks appear in many cones: diffusion is a random walk process undersanding buffering, waiing imes, queuing more generally he heory of sochasic processes gambling choosing he bes

### Hedging with Forwards and Futures

Hedging wih orwards and uures Hedging in mos cases is sraighforward. You plan o buy 10,000 barrels of oil in six monhs and you wish o eliminae he price risk. If you ake he buy-side of a forward/fuures

### PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE

Profi Tes Modelling in Life Assurance Using Spreadshees PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Erik Alm Peer Millingon 2004 Profi Tes Modelling in Life Assurance Using Spreadshees

### Two Compartment Body Model and V d Terms by Jeff Stark

Two Comparmen Body Model and V d Terms by Jeff Sark In a one-comparmen model, we make wo imporan assumpions: (1) Linear pharmacokineics - By his, we mean ha eliminaion is firs order and ha pharmacokineic

### 11. Tire pressure. Here we always work with relative pressure. That s what everybody always does.

11. Tire pressure. The graph You have a hole in your ire. You pump i up o P=400 kilopascals (kpa) and over he nex few hours i goes down ill he ire is quie fla. Draw wha you hink he graph of ire pressure

### Chapter 6 Interest Rates and Bond Valuation

Chaper 6 Ineres Raes and Bond Valuaion Definiion and Descripion of Bonds Long-erm deb-loosely, bonds wih a mauriy of one year or more Shor-erm deb-less han a year o mauriy, also called unfunded deb Bond-sricly

### Rotational Inertia of a Point Mass

Roaional Ineria of a Poin Mass Saddleback College Physics Deparmen, adaped from PASCO Scienific PURPOSE The purpose of his experimen is o find he roaional ineria of a poin experimenally and o verify ha

### Risk Modelling of Collateralised Lending

Risk Modelling of Collaeralised Lending Dae: 4-11-2008 Number: 8/18 Inroducion This noe explains how i is possible o handle collaeralised lending wihin Risk Conroller. The approach draws on he faciliies

### Answer, Key Homework 2 David McIntyre 45123 Mar 25, 2004 1

Answer, Key Homework 2 Daid McInyre 4123 Mar 2, 2004 1 This prin-ou should hae 1 quesions. Muliple-choice quesions may coninue on he ne column or page find all choices before making your selecion. The

### Lecture III: Finish Discounted Value Formulation

Lecure III: Finish Discouned Value Formulaion I. Inernal Rae of Reurn A. Formally defined: Inernal Rae of Reurn is ha ineres rae which reduces he ne presen value of an invesmen o zero.. Finding he inernal

### 1. The graph shows the variation with time t of the velocity v of an object.

1. he graph shows he variaion wih ime of he velociy v of an objec. v Which one of he following graphs bes represens he variaion wih ime of he acceleraion a of he objec? A. a B. a C. a D. a 2. A ball, iniially

### Stability. Coefficients may change over time. Evolution of the economy Policy changes

Sabiliy Coefficiens may change over ime Evoluion of he economy Policy changes Time Varying Parameers y = α + x β + Coefficiens depend on he ime period If he coefficiens vary randomly and are unpredicable,

### Inductance and Transient Circuits

Chaper H Inducance and Transien Circuis Blinn College - Physics 2426 - Terry Honan As a consequence of Faraday's law a changing curren hrough one coil induces an EMF in anoher coil; his is known as muual

### SPEC model selection algorithm for ARCH models: an options pricing evaluation framework

Applied Financial Economics Leers, 2008, 4, 419 423 SEC model selecion algorihm for ARCH models: an opions pricing evaluaion framework Savros Degiannakis a, * and Evdokia Xekalaki a,b a Deparmen of Saisics,

### Week #9 - The Integral Section 5.1

Week #9 - The Inegral Secion 5.1 From Calculus, Single Variable by Hughes-Halle, Gleason, McCallum e. al. Copyrigh 005 by John Wiley & Sons, Inc. This maerial is used by permission of John Wiley & Sons,

### Chapter 1 Overview of Time Series

Chaper 1 Overview of Time Series 1.1 Inroducion 1 1.2 Analysis Mehods and SAS/ETS Sofware 2 1.2.1 Opions 2 1.2.2 How SAS/ETS Sofware Procedures Inerrelae 4 1.3 Simple Models: Regression 6 1.3.1 Linear

### 13 Solving nonhomogeneous equations: Variation of the constants method

13 Solving nonhomogeneous equaions: Variaion of he consans meho We are sill solving Ly = f, (1 where L is a linear ifferenial operaor wih consan coefficiens an f is a given funcion Togeher (1 is a linear

### Convexity. Concepts and Buzzwords. Dollar Convexity Convexity. Curvature, Taylor series, Barbell, Bullet. Convexity 1

Deb Insrumens and Markes Professor Carpener Convexiy Conceps and Buzzwords Dollar Convexiy Convexiy Curvaure, Taylor series, Barbell, Bulle Convexiy Deb Insrumens and Markes Professor Carpener Readings

### Cointegration Analysis of Exchange Rate in Foreign Exchange Market

Coinegraion Analysis of Exchange Rae in Foreign Exchange Marke Wang Jian, Wang Shu-li School of Economics, Wuhan Universiy of Technology, P.R.China, 430074 Absrac: This paper educed ha he series of exchange

### RC (Resistor-Capacitor) Circuits. AP Physics C

(Resisor-Capacior Circuis AP Physics C Circui Iniial Condiions An circui is one where you have a capacior and resisor in he same circui. Suppose we have he following circui: Iniially, he capacior is UNCHARGED

### A Re-examination of the Joint Mortality Functions

Norh merican cuarial Journal Volume 6, Number 1, p.166-170 (2002) Re-eaminaion of he Join Morali Funcions bsrac. Heekung Youn, rkad Shemakin, Edwin Herman Universi of S. Thomas, Sain Paul, MN, US Morali

### INTRODUCTION TO FORECASTING

INTRODUCTION TO FORECASTING INTRODUCTION: Wha is a forecas? Why do managers need o forecas? A forecas is an esimae of uncerain fuure evens (lierally, o "cas forward" by exrapolaing from pas and curren

### MACROECONOMIC FORECASTS AT THE MOF A LOOK INTO THE REAR VIEW MIRROR

MACROECONOMIC FORECASTS AT THE MOF A LOOK INTO THE REAR VIEW MIRROR The firs experimenal publicaion, which summarised pas and expeced fuure developmen of basic economic indicaors, was published by he Minisry

### What is a Trait? What is Genetics? What are Genes? What is an Allele? Mendel s Conclusions. Gregor Mendel 1/11/2016

Wha is Geneics? Geneics is he scienific sudy of herediy Wha is a rai? A rai is a specific characerisic ha varies from one individual o anoher. Examples: Brown hair, blue eyes, all, curly Wha is an Allele?

### Table of contents Chapter 1 Interest rates and factors Chapter 2 Level annuities Chapter 3 Varying annuities

Table of conens Chaper 1 Ineres raes and facors 1 1.1 Ineres 2 1.2 Simple ineres 4 1.3 Compound ineres 6 1.4 Accumulaed value 10 1.5 Presen value 11 1.6 Rae of discoun 13 1.7 Consan force of ineres 17

### Differential Equations. Solving for Impulse Response. Linear systems are often described using differential equations.

Differenial Equaions Linear sysems are ofen described using differenial equaions. For example: d 2 y d 2 + 5dy + 6y f() d where f() is he inpu o he sysem and y() is he oupu. We know how o solve for y given

### Advanced time-series analysis (University of Lund, Economic History Department)

Advanced ime-series analysis (Universiy of Lund, Economic Hisory Deparmen) 30 Jan-3 February and 6-30 March 01 Lecure Uni-roo esing and he consequences of non-saionariy on regression analysis..a. Why is

### Differential Equations and Linear Superposition

Differenial Equaions and Linear Superposiion Basic Idea: Provide soluion in closed form Like Inegraion, no general soluions in closed form Order of equaion: highes derivaive in equaion e.g. dy d dy 2 y