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Financial Econometrics – Slides-06: Generalizing to ARMA and Forecasting

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Dr. School of Economics (UNSW) Slides-06 ©UNSW 1 / 33

Financial Econometrics
Slides-06: Generalizing to ARMA and Forecasting

School of Economics1

1©Copyright University of Wales 2020. All rights reserved. This copyright notice must not
be removed from this material.

Dr. School of Economics (UNSW) Slides-06 ©UNSW 2 / 33

• General AR(p)
• Wold Decomposition
• AF and PACF patterns
• Impulse response function

• Yule-Walker equations
• AR & MA mix- ARMA models

• AF and PACF patterns
• Impulse response function

• Estimation of ARMA
• Forecasting in ARMA

Dr. School of Economics (UNSW) Slides-06 ©UNSW 3 / 33

Stationarity of AR(2)

Univariate Time Series Analysis: ARIMA models

Building ARIMA models

Autoregressive Process

The conditions for stationarity/invertibility of an AR(1) process
can be extended to higher order AR processes.

I First consider an AR(2) process

1 � ↵1L � ↵2L2

yt = ↵ (L) yt = ↵0 + “t .

In general, the polynomial ↵ (L) can be rewritten as

1 � ↵1L � ↵2L2

= (1 � �1L) (1 � �2L) .

where �1 and �2 can be solved from �1 + �2 = ↵1 and
��1�2 = ↵2

The conditions for invertibility of the second order polynomial
are just the conditions that both the first order polynomials
(1 � �1L) and (1 � �2L) are invertible, i.e. |�1| < 1 and Dr. School of Economics (UNSW) Slides-06 ©UNSW 4 / 33 Stationarity of AR(2) ©Copyright University of Wales 2020. All rights reserved. This copyright notice must not be removed from this material Univariate Time Series Analysis: ARIMA models Building ARIMA models Autoregressive Process A more common way of presenting these conditions is in terms of the so-called characteristic equation 1 � ↵1z � ↵2z2 or (1 � �1z) (1 � �2z) = 0. This equation has two solutions, denoted z1 and z2 referred to as the characteristic roots of the ↵ (L) polynomial. The requirement |�i | < 1 corresponds to |zi | > 1. If any solution
satisfies |zi |  1, the polynomial ↵ (L) is non-invertible. A solution
that equals unity is referred to as a unit root.

General Conditions for Stationarity for an AR(p)

Univariate Time Series Analysis: ARIMA models

Building ARIMA models

Autoregressive Process

Calculating the roots of a higher order AR process is
computationally not a trivial job. In most circumstances there is
little need to directly calculate the characteristic roots, though, as
there are some useful simple rules for checking
stationarity/invertibility of higher order processes

I Necessary condition:

i=1 ↵i < 1 I Su�cient condition: i=1 |↵i | < 1 Dr. School of Economics (UNSW) Slides-06 ©UNSW 6 / 33 Useful representations ©Copyright University of Wales 2020. All rights reserved. This copyright notice must not be removed from this material Univariate Time Series Analysis: ARIMA models Building ARIMA models Autoregressive Process As, under appropriate conditions, an AR(p) process has an MA(1) representation and an MA(q) has an AR(1) representation, there is no fundamental di↵erence between AR and MA models. I The MA representation is convenient to derive the properties (mean, variance, ...) of a series I The AR representation is convenient for making predictions conditional upon the past When estimating time series models (cf. below), the choice is simply a matter of parsimony. Dr. School of Economics (UNSW) Slides-06 ©UNSW 7 / 33 What is the AR process is stationary? ©Copyright University of Wales 2020. All rights reserved. This copyright notice must not be removed from this material Univariate Time Series Analysis: ARIMA models Building ARIMA models Autoregressive Process For a stationary AR(p) process, it is more convenient to derive the properties from imposing that the mean, variance and autocovariances do not depend on time. For computational convenience consider an AR(2) process. I The unconditional mean of yt can be solved from E (yt) = ↵0 + ↵1E (yt�1) + ↵2E (yt�2) which, assuming that E (yt) does not depend on time allows us to write E (yt) = ↵0 /(1 � ↵1 � ↵2) I The variance of yt can be solved by defining xt = yt � E (yt) which yields xt = ↵1xt�1 + ↵2xt�2 + "t (18) Dr. School of Economics (UNSW) Slides-06 ©UNSW 8 / 33 What is the AR process is stationary? ©Copyright University of Wales 2020. All rights reserved. This copyright notice must not be removed from this material Univariate Time Series Analysis: ARIMA models Building ARIMA models Autoregressive Process The variance of yt can be obtained by multiplying both sides by xt and taking expectations V (yt) = �0 = E (↵1xtxt�1 + ↵2xtxt�2 + xt"t) = ↵1�1 + ↵2�2 + E (xt"t) = ↵1�1 + ↵2�2 + � where E (xt"t) = � 2 is obtained from multiplying both sides of (18) by "t and taking expectations. Multiplying both sides by xt�1 and xt�2 and taking expectations we obtain �1 = ↵1�0 + ↵2�1 (20) �2 = ↵1�1 + ↵2�0 (21) These equations can be solved for �0 to obtain (1 + ↵2) (1 � ↵1 � ↵2) (1 + ↵1 � ↵2) Dr. School of Economics (UNSW) Slides-06 ©UNSW 9 / 33 What is the AR process is stationary? ©Copyright University of Wales 2020. All rights reserved. This copyright notice must not be removed from this material Univariate Time Series Analysis: ARIMA models Building ARIMA models Autoregressive Process I The autocorrelation coe�cients ⇢1 and ⇢2 can be obtained by dividing (20) and (21) by �0 ⇢1 = ↵1 + ↵2⇢1 ⇢2 = ↵1⇢1 + ↵2 and solving to obtain ⇢1 = ↵1 /(1 � ↵2) 1 /(1 � ↵2) + ↵2 It is easily verified that the higher-order autocorrelation coe�cients are given by ⇢k = ↵1⇢k�1 + ↵2⇢k�2 Dr. School of Economics (UNSW) Slides-06 ©UNSW 10 / 33 ©Copyright University of Wales 2020. All rights reserved. This copyright notice must not be removed from this material � The beauty of the yule Walker Equations! xt = α1xt−1 + · · ·+ αpxt−1 + �t xtxt−1 = α1xt−1xt−1 + · · ·+ αpxt−pxt−1 + �txt−1 E(xtxt−1) = α1E(xt−1xt−1) + · · ·+ αpE(xt−pxt−1) + E(�txt−1) γ1 = α1γ0 + α2γ1 + · · ·+ αpγp−1 xtxt−j = α1xt−1xt−j + · · ·+ αpxt−pxt−j + �txt−j E(xtxt−j) = α1E(xt−1xt−j) + · · ·+ αpE(xt−pxt−j) + E(�txt−j) γ|j| = α1γ|j−1| + α2γ|j−2| + · · ·+ αpγ|p−j|, Dr. School of Economics (UNSW) Slides-06 ©UNSW 11 / 33 Defining an ARMA Process ©Copyright University of Wales 2020. All rights reserved. This copyright notice must not be removed from this material Univariate Time Series Analysis: ARIMA models Building ARIMA models An ARMA Process Defining an ARMA Process Let "t be a white noise process. Then: ↵ (L) yt = ↵0 + � (L) "t (22) with ↵ (L) an AR polynomial of order p and � (L) an MA polynomial of order q, is an autoregressive moving average process with orders p and q, denoted ARMA(p, q). ! yt depends on its own lagged values and on current and past values of a white noise disturbance term "t . Dr. School of Economics (UNSW) Slides-06 ©UNSW 12 / 33 Dynamic Behaviour ©Copyright University of Wales 2020. All rights reserved. This copyright notice must not be removed from this material � Dynamic Behaviour and Impulse Response Univariate Time Series Analysis: ARIMA models Building ARIMA models An ARMA Process Dynamic Behaviour of an ARMA(p, q) Process If the AR polynomial ↵ (L) is invertible, the ARMA(p, q) process can be written as a stable MA(1) process of the form yt = ↵ (L) ↵0 + ↵ (L) = ↵00 + ✓ (L) "t where ↵00 = ↵0 i=1 ↵i and ✓ (L) = ↵ (L) i , with ✓i = undetermined coe�cients. Even if the AR polynomial is non-invertible, we can still solve for the " sequence but this solution will not be a stable MA process, yt = f (t) + ✓ (L) "t where f (t) indicates that the mean is a function of time. Dr. School of Economics (UNSW) Slides-06 ©UNSW 13 / 33 Dynamic Behaviour ©Copyright University of Wales 2020. All rights reserved. This copyright notice must not be removed from this material � Dynamic Behaviour and Impulse Response Univariate Time Series Analysis: ARIMA models Building ARIMA models An ARMA Process The impulse response function can be obtained from the MA representation. Note that as a finite order MA process is stationary by construction, an ARMA process is stationary if the AR component is stationary (i.e. if the AR polynomial is invertible). I In the stationary case the impact of shocks gradually dies out i=1 ✓i is finite) I In the non-stationary case the impact of a shock never vanishes (i.e. i=1 ✓i is infinite) Dr. School of Economics (UNSW) Slides-06 ©UNSW 14 / 33 General Properties of an ARMA(p,q) ©Copyright University of Wales 2020. All rights reserved. This copyright notice must not be removed from this material � Unconditional Moments of an ARMA(p, q) Univariate Time Series Analysis: ARIMA models Building ARIMA models An ARMA Process Properties of an ARMA(p, q) Process If the AR polynomial ↵ (L) is non-invertible the mean, variance and covariances are time-varying. If the AR polynomial ↵ (L) is invertible, the AR process can be rewritten as the stable infinite MA process. The properties of a stationary AR process can easily be derived from this MA representation. I Unconditional mean: E (yt) = ↵0 I Unconditional variance: V (yt) = � I Covariances: �k = (✓k + ✓1✓k+1 + ✓2✓k+2 + . . .)� As an ARMA(p, q) process includes both an AR and an MA component, both the ACF and the PACF do not cut o↵ at some point. As such, it is di�cult to determine the order of an ARMA model from the ACF and PACF. Dr. School of Economics (UNSW) Slides-06 ©UNSW 15 / 33 Maximum Likelihood Estimation: Intuitive Illustration ©Copyright University of Wales 2020. All rights reserved. This copyright notice must not be removed from this material Maximum Likelihood Estimation Binary dependent variable model The Probit Model Inference in Logit and Probit Specification tests Intuitive illustration This illustration shows a sample of n independent observations, and two continuous distributions f1(x) and f2(x), Likelihood This illustration shows a sample of n independent observations, and two continuous distributions f1(x) and f2(x), with f2(x) being just f1(x) translated by a certain amount. Of these two distributions, which one is the most likely to have generated the sample ? Clearly, the answer is f1(x), and we would like to formalize this intuition. Although this is not strictly impossible, we don't believe that f2(x) generated the sample because all the observations are in regions where the values of f2(x) are small : the probability for an observation to appear in such a region is small, and it is even more unlikely that all the observations in the sample would appear in low density regions. On the other hand, the values taken by f1(x) are substantial for all the observations, which are then where one would expect them to be, would the sample be actually generated by f1(x). Definition of the likelihood Of the many ways to quantify this intuitive judgement, one turns out to be remarkably effective. For any probability distribution f(x), just multiply the values of f(x) for each of the observations of the sample, denote the result L, and call it the likelihood of the distribution f(x) for this particular sample : Clearly, the likelihood can have a large value only if all the observations are in regions where f(x) is not very small. This definition has the additional advantage that L receives a natural interpretation. The sample {xi} may be regarded as a single observation generated by the n-variate probability distribution f(x1, x2, ..., xn) = Πi f(xi) because of the independence of the individual observations. So the likelihood of the distribution is just the value of the n-variate probability density f (x1, x2, ..., xn ) for the set of observations in the sample considered as a unique n-variate observation. Likelihood and estimation, Maximum Likelihood estimators These considerations make us believe that "likelihood" might be a helpful concept for identifying the distribution that generated a given sample. First note, though, that as such, this approach is moot if we don't a priori restrict our search : the probability distribution leading to the largest possible value of the likelihood is obtained by assigning the probability 1/n to each of the points where there is an observation, and assigning the value 0 to f(x) for any other point of the x axis. This result is both trivial and useless. But consider the example given in the above illustration : f1(x) and f2(x) are assumed to belong to a family of distributions, all identical in shape and differing only by their position along the x axis (location family). It now makes sense to ask for which position of the generic distribution f(x) is the likelihood largest. If we denote θ the parameter adjusting the horizontal position of the distribution, one may consider the value of θ conducive to the largest likelihood as being probably fairly close to the true (and unknown) value θ of the parameter of the distribution that actually generated the It then appears that the concept of likelihood may lead to a method of parameter estimation. The method consists in retaining as an estimate of θ value of θ conducive to the largest possible value of the sample likelihood. This method is thus called Maximum Likelihood estimation, which is, in fact, the most powerful and widely used method of parameter estimation these days. An estimator θ* obtained by maximizing the likelihood of a probability distribution defined up to the value of a parameter θ is called a Maximum Likelihood estimator and is usually denoted "MLE". When we need to emphasize the fact that the likelihood depends on both the sample x = {xi} and the parameter θ, we'll denote it L(x, θ). Interactive animation Likelihood = L =: Πi f(xi) i = 1, 2, ..., n Page 1 of 6Likelihood and method of Maximum Likelihood 21/03/2010http://www.aiaccess.net/English/Glossaries/GlosMod/e_gm_likelihood.htm Of these two distributions, which one is the most likely to have generated the sample ? Although it is not impossible, we don’t believe that f2(x) generated the sample. Why? On the other hand, the values taken by f1(x) are substantial for all the observations, which are then where one would expect them to be, would the sample be actually generated by f1(x). Dr. ECON3208: Lecture 3 Maximum Likelihood Estimation Limited Dependent Variable Models Dr. School of Economics (UNSW) Slides-06 ©UNSW 16 / 33 Maximum Likelihood Estimation ©Copyright University of Wales 2020. All rights reserved. This copyright notice must not be removed from this material Maximum Likelihood Estimation Binary dependent variable model The Probit Model Inference in Logit and Probit Specification tests Maximum Likelihood Estimation • Maximum Likelihood Estimation is a general method of estimation that can be used for many di↵erent types of data and economic models. It has very wide applicability. • The Maximum Likelihood Estimator (MLE) answers the following question: What are the parameter estimates that are most likely to have generated the observed data given the assumed model. • Begin by assuming a model for the outcome variable including a distribution function for the underlying population error term (and hence a distribution for the outcome variable in the population.) Dr. ECON3208: Lecture 3 Maximum Likelihood Estimation Limited Dependent Variable Models Dr. School of Economics (UNSW) Slides-06 ©UNSW 17 / 33 Estimation of ARMA: Maximum Likelihood ©Copyright University of Wales 2020. All rights reserved. This copyright notice must not be removed from this material • Consider AR(1) model: yt = α0 + α1yt−1 + �t where �t ∼ i.i.d N(0, σ2). - it follows: yt|Ωt−1 ∼ N α0 + α1yt−1, σ , t = 2, 3, · · · . [1− α1]−1α0, [1− α21] - Conditional pdf: f(yt|Ωt−1) = (yt − α0 − α1yt−1)2 - Information sets: Ω1 = {y1},Ω2 = {y2,Ω1}, · · · ,Ωt = {yt,Ωt−1}. - Joint pdf for a time series {y1, · · · , yT } can be factorised: f(yT , yT−1 · · · , y1) = = f(yT , yT−1 · · · , y2|Ω1)f(y1) = f(yT , yT−1 · · · , y3|Ω2)f(y2|Ω1)f(y1) = f(yT |ΩT−1)f(yT−1|ΩT−2) · · · f(y3|Ω2)f(y2|Ω1)f(y1) Dr. School of Economics (UNSW) Slides-06 ©UNSW 18 / 33 Maximum Likelihood Estimation ©Copyright University of Wales 2020. All rights reserved. This copyright notice must not be removed from this material Topic 3. Time Series Models • Maximum likelihood – Properties of ML estimators • When the pdf (likelihood) is correctly specified, the ML estimators have nice large-𝑇𝑇 sampling properties: – consistent, – asymptotically normally distributed, and – asymptotically efficient. Allow us to draw inference based on reported SEs. • When the pdf (likelihood) is incorrect, the “ML” procedure is called quasi (or pseudo) ML. – When the normal pdf is used, which may be incorrect, the quasi ML estimators are still consistent and asymptotically normal, as long as the model is defined by the conditional mean and variance that are correctly specified. School of Economics, UNSW Slides-05, Financial Econometrics 14 “robust” SEs Dr. School of Economics (UNSW) Slides-06 ©UNSW 19 / 33 ARMA Process: Identification ©Copyright University of Wales 2020. All rights reserved. This copyright notice must not be removed from this material Univariate Time Series Analysis: ARIMA models Fitting ARMA models to the data The so-called Box-Jenkins approach toward fitting ARMA models comprises three stages: I Identification: determine tentative model(s) I Plot the time series to have a first idea on the DGP (stationary/non-stationary, structural break, ...) I Plot the ACF and the PACF to have a first idea on the order of the ARMA model I Estimation: estimate the various tentative models I Compare the estimated models using information criteria I Select parsimonious model I Diagnosti 程序代写 CS代考加微信: powcoder QQ: 1823890830 Email: powcoder@163.com

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