Time Series Modeling Computation And Inference 2nd Edition Raquel Prado, Marco A. R. Ferreira, Mike West - Solutions

Discover comprehensive solutions for "Time Series Modeling Computation and Inference, 2nd Edition" by Raquel Prado, Marco A. R. Ferreira, and Mike West. Access an extensive collection of questions and answers, featuring an online answers key and a detailed solution manual. This resource offers solutions in PDF format, including solved problems and step-by-step answers, ensuring a deep understanding of the textbook. Enhance your learning with the test bank and chapter solutions, ideal for instructors and students alike. Explore the instructor manual and enjoy a free download of this invaluable educational tool.

Consider the k-factor model given in Section 11.2.4.(a) Simulate a sample of 200 observations of 15-dimensional vectors with k = 3 factors assuming 2 j = 01 +002 j j = 1 12, h1 = 15, h2 = 05, h3 =025, and 2 =036.(b) Implement the MCMC algorithm presented in Section 11.2.5 to ex plore the posterior
In the context of selection of number of factors of Section 11.2.6, show that the integrated likelihood function for the parameters of the k-factor model obtained by integrating out the latent factor vectors x1 xn is given by (Lopes and West 2004)p(yk k)=(2 ) nr 2BHB+V n2exp 1 2n i=1 yi(BHB+V) 1yi
Consider the one-factor model given by Equation (11.1).(a) Simulate a sample of 200 observations of 12-dimensional vectors as suming 2 j = 01+001 j j =1 12, H =1, and 2 =036.(b) Implement the MCMC algorithm given in Section 11.2.2 to explore the posterior distribution.(c) Analyze the simulated data
Consider the k-factor model presented in Section 11.2.4. Derive the full conditional distributions for the unknown model quantities given in Sec tion 11.2.5.
Consider the k-factor model given by Equation (11.8). Show that if no constraints are imposed on B or xi then the model is not identi able.
Assume the one-factor model given by Equation (11.1). Derive the full conditional distributions for the unknown model quantities given in Sec tion 11.2.2.
VerifytheWishartevolutiondistributiontheoryofSection10.4.8inthe followinggeneralsetting.Theq qprecisionmatrix hastheWishartdistribution W(hA)for somedegreesof freedomh=n+q 1wheren>0sothath>q 1 andwhereAisthe inversesum-of-squaresmatrix.TheBartlett
Verify the optimal portfolio construction theory for one-step-ahead port folio decisions summarized in Section 10.4.7. That is, suppose that the predictive moments ft = E(yt Dt 1) and Vt = V(yt Dt 1) are available, with forecast precision matrix Kt = V1 tproblem is to nd wt such that The
Develop software to implement the forward ltering/sequential updat ing analysis, and the retrospective smoothing computations, for the ex changeable time series DLM with time-varying observational covariance matrix under the matrix beta evolution model. Explore the exchange rate time series data in
Use the retrospective ltering analysis of Equation (10.12) to derive a retrospective recursive equation for computing the sequence of estimates E( t DT) over t = (T 1): 1
Simplify the retrospective ltering of Equation (10.12) for the case of a univariate time series, q = 1Describe how the retrospectively simulated values of past time-varying precisions now depend on a sequence of ran dom retrospective shocks that have 2 ht distributions (t = (T 1) : 1)
Consider two independent q q Wishart matrices S1 W( 1A) and S2 W( 2A) where 1 = h and 2 = (1 )h for some h > q 1 and(01)What is the distribution of S = S1 +S2?Use Corollary 2 of Dawid (1981) to show that S1 = U U where U satis es S = UU and has the matrix beta distribution Be( h2 (1 )h2)Use the
In the exchangeable time series model of Section 10.3, suppose that Gt = Ip for all t and that the Wt sequence is de ned via a single discount factor so Rt = Ct 1 for all t This models evolution of all state parameters as steady models, or random walks, evolving at a constant discount rate. Show
Supposetheq vectoryt=(yt1 ytq) timeseries isVARq(1)with yt= yt 1+ tand t N(0V) independentlyovertime.(a)Takeanynonsingularq qmatrixAandconsiderthetransformation toxt=Ayt ShowthatthisyieldsaVARq(1)processxtandgivethe expressions fortheVARcoe cientmatrixandinnovationsvariance matrix.(b)
Supposetheq vectortimeseriesyt followsaVARq(p)modelwith yt=p r=1 ryt r+ t t N(0V)withindependent innovations tovertime.(a) Showthat themodel canbewrittenasyt = Ft+ t for some regressionvectorFtandmatrixparameter .Givethede nitionsof Ft and andcommentontheirdimensions.Thisresultshowsthat
Two stationary, univariate AR(1) processes are driven by correlated in novation sequences. That is, we observe two processes yt and zt where yt= yt 1 + t t N(0v)zt = zt 1+ t t N(0w)where, as usual, t and t are independent over time, i.e., t and t are independent of t k and of t k for any k >
Consider a simple VAR2(1) model for xt = (xt1 xt2) given by xt =Gxt 1+ t with t =( t1 t2) N(0W)Denotethe(i j) element of G by gij so the evolution equation element-by-element is xt1 = g11xt 11 +g12xt 12 + t1 xt2 = g21xt 11 +g22xt 12 + t2(a) Write these equations in terms of backshift operators and
Simulatethree-dimensionaldatayt fromthefollowingmodel:yt =1 1 0 1 0 1 1 1 0 xt+ t with t N(04I3)andxt=(x1t x2t x3t)suchthat x1t = 095x1t 1+ 1t x2t = 2 095 cos(2 18)x2t 1 0952x2t 2+ 2t x3t = 2 095 cos(2 7)x3t 1 0952x3t 2+ 3t where t=( 1t 2t 3t) and t N(0I3) Inotherwords,x1t isan AR(1)processwithcoe
ShowthedecompositionresultsforVARmodelssummarizedinSection 9.1.3.
Sketch the MCMC algorithm for obtaining samples from the joint pos terior distribution in the dynamical lag/lead model presented in Section 8.3
Find the phase spectrum for the process in (8.10).
Show that if yt1 N(01) for all t and yt2 is given by yt2 = 1 3 (yt 11 +yt1 +yt+11)the coherency of the process yt = (yt1 yt2) is zero.
Consider the two-dimensional process yt = (yt1 yt2) de ned as yt1 = t1 yt2 = yt+d1 + t2 where ti are independent, mutually independent, zero-mean processes with ti N(01) Find the cross-spectrum, the amplitude, and phase spectra, as well as the squared coherency of yt1 and yt2.
Show that the bivariate process de ned by (8.10) has the spectrum given in (8.11) and therefore the implied coherency of the process is one.
Show that the coherency can be written in terms of the amplitude and phase spectra, 12( ) and 12( ) as 12( )[f11( )f22( )]1 2 exp i 12( )
Show that the cospectrum and the quadrature spectrum de ned in (8.7)can be written as c12( ) = 1 2 h=12(h)cos(h )C12( ) =q12( ) = 1 2 h=12(h)sin(h )In addition, show that c12( ) = c21( ) and q12( ) = q21( )
Work out the details of the Gibbs sampling algorithm sketched in Sec tion 8.1 for obtaining samples from the posterior distribution of the parameters in model (8.1). In particular, show that sampling from the full conditional distributions in Steps 1 and 2 can be achieved via FFBS.
Simulate data from the following model:yt = (1)yt 1+ t t=1:200 yt = (2)yt 1+ t t=201:400 where (1) = 09 (2) = 09 and t N(0v) with v = 1 Assuming priors of the form (1) TN(051R1) and (2) TN( 051R2) with R1 =(01) and R2 =( 10)ontheARcoe cients and an inverse-gamma prior on v compute Pr(Mt(1) Dt) with
A stationary, rst order Markov process is generated from p(xtxt 1)given by xt where N(xt xt 1 v) with probability N(xt0s)otherwise,
In the univariate SV model assuming the normal mixture form of p( t)what is the stationary marginal distribution p(yt v) for any t implied by this model? Investment analysis focuses, in part, on questions of just how large or small a per-period return might be, and this marginal distribution of a
An econometrician notes that the assumed normal mixture error distri bution used in the SV model analysis is just an approximation to the sampling distribution of the data, and is worried that the speci c val ues of the (qj bj wj) might be varied to better t the data in a speci c application. To
In the SV model context, derive expressions for the parameters of the component distributions for the MCMC analysis of Section 7.5.4 includ ing the expressions for the sequences of conditional normal distributions p(ztzt+1 y1:t 1:t v) t = (T 1) : 0 for the FFBS algorithm to simulate z0:T Implement
Generate a reasonably large random sample from the 2 1 distribution and look at histograms of the implied values of t = log( t) 2 where 2t 1 as in the SV model. Explore the shape of this distribution using quantile-quantile plots against the standard normal distribution to generate insight into how
Suppose that xt 1 N(xt 10s) and xt is generated by xt = ztxt 1 +(1 zt) t where t N(0s) and zt is binary with Pr(zt = 1) = a and with xt 1 zt t being mutually independent. Here sa are known.(a) What is the distribution of xt?(b) What is the correlation C(xt xt 1)?(c) Simulate and graph realizations
Consider the logistic DLM given by (see Storvik 2002)yt tBin(r logit( + t))N( t 1w)Simulate T = 300 observations from this model with = 09, w = 1 and = =05(a) Assume that and are known. Implement a SMC algorithm for sequential ltering and parameter learning of and w assuming a Gaussian prior on and
Consider the AR(1) plus noise model in Example 6.6.(a) Run Liu and Wests algorithm with di erent discount factors.(b) Implement the algorithm of Liu and West (2001) using an importance density that conditions on yt p( t t 1yt) instead of one that does not condition on yt(c) Implement the algorithm
Consider the fat-tailed nonlinear state-space model studied in Carvalho, Johannes, Lopes, and Polson (2010) and given by yt = t+v t t t=t 1(1 + 2 t 1) +w t where t N(01) t N(01)and t IG( 2 2)(a) Assume that vwand areknown. Propose, implement, and com pare SMC approaches for ltering without
Consider the PACF TVAR(2) parameterization in Example 5.4. Simu late data x1:T from this model.(a) Sketch and implement a SMC algorithm for ltering and smoothing assuming that v and w are known.(b) Sketch and implement a SMC algorithm for ltering and smoothing assuming that v and w are unknown.
Consider the AR(1) state plus noise model discussed in Example 6.6.Implement the algorithm of Storvik (2002) for ltering with parameter estimation and compare it to other SMC algorithms such as that of Liu and West (2001) and the PL algorithm of Carvalho, Johannes, Lopes, and Polson (2010)
Consider the dynamic trend model FGv( 1)W( 2 3) introduced by Harrison and Stevens (1976) and revisited in Fruhwirth-Schnatter(1994), where F=(10) G= 1 1 0 1 v( 1)= 1 and W( 2 3)=Gdiag( 2 3)G= 2+ 3 3 3 3 Simulate a time series data set from this model. Propose and implement a MCMC algorithm for
Consider again the AR(1) model with mixture observational errors de scribed in Example 4.10. Modify the MCMC algorithm in order to per form posterior inference when t has the following Markovian structure:Pr( t = 2 t 1= 2)=Pr( t=1 t 1=1)=p and Pr( t = 2 t 1=1)=Pr( t=1 t 1= 2)=(1 p)where p is known.
Derive the conditional distributions for posterior MCMC simulation in Example 4.10, verifying that the algorithm outlined there is correct.
Consider the following model:yt = t+ t t N(0 2)p t=j t j + t t N(0 2)j=1 This model is a simpli ed version of that proposed in West (1997c).Develop the conditional distributions required to de ne an MCMC algo rithm to obtain samples from p( 1:T 2 2y1:T) and implement the algorithm.
Go to Google Trends and download the monthly data for the searches of the term time series in the U.S. and the rest of the world from January 2004 until December 2019. For each of these two time series t the following DLMs and provide a summary of the ltering and smoothing distributions of the
Consider the observational variance discount model of Section 4.3.7. You may use the results from Problem 10.(a) Show that the time t 1 prior ( t 1 Dt 1) G(nt 1 2dt 1 2) com bined with the beta-gamma evolution model t = t 1 t yields a conditional density p( t 1 tDt 1) that can be expressed as t 1
The basic distribution theory in this question underlies the discount volatility model of Section 4.3.7 and the results to be shown below in Problem 11. Two positive scalar random quantities 0 and 1 have a joint distribution under which:0 G(ab) for some scalars a > 0b > 0; and p( 1 0) is implicitly
Work through the key results of Section 4.3.5 to ensure understanding of the role of the Markovian structure of a DLM in retrospective analysis.Do this in a DLM which, for all time t, has known observation variance vt Given Dt 1 the two consecutive state vectors t and t 1 are related linearly with
Consider the three DLMs below, each with a 2 dimensional state vector t = ( t1 t2). Each model is de ned by the constant FG elements shown. For each of these DLMs, give details of the implied form of the forecast function ft(k) over k =12 and comment on the meaning/interpretation of the elements of
A DLM has the forecast function de ned over k = 01 current time t by ft(k) = at1 + at2k + at3rk cos(2 k +ct)for some positive wavelength and some positive number r < 1, and where the quantities at1 at2 at3 ct are constants known at time t. Give real-valued and constant observation vector F and
A DLM for the univariate series yt is given by yt = F t + t where tN(0v), and t = Gt 1+ twhere t N(0vW)withtheusual conditional independence assumptions. All model parameters FvGW are known and constant over time. The modeler speci es the model such that:G has p real and distinct eigenvalues i i =
Consider a dynamic regression DLM for a univariate time series, namely yt = Ft t+ t with t N(0v) and v known. Suppose a random walk evolution for t so that G = I and t = t 1+ t and t N(0vWt)where Wt is de ned by a single discount factor With an initial prior 0 D0 N(m0 vC0) it follows for all t 1
For a univariate series yt consider the simple rst-order polynomial (lo cally constant) DLM with local level t at time t. The p = 1 dimensional state is t = t, while Ft = 1 and Gt = 1 for all t Also, assume a con stant, known observation variance v(a) Show that the usual updating equations for mtCt
Show that the smoothing equations in (4.10) and (4.11) can be written(01] is used to 1we have the summary posterior ( t 1 Dt 1) N(mt 1Ct 1) and the state vector evolves through the state equation t = Gt t 1 + t where t N(0Wt) with t 1 t are independent. Suppose now the special case in which:the
Assuming a DLM structure given by FtGt vtWt , nd the distribu tions of ( t+k t+j Dt), (yt+k yt+j Dt), ( t+k yt+j Dt) (yt+k t+j Dt)and ( t k j t k Dt).
Let xt be a stationary AR(1) process given by xt = xt 1 + x t with N(0vx) Let yt = xt + y t with y t uncorrelated with the process xt and with y t N(0vy)(a) Find the spectrum of yt .(b) Simulate 500 observations from the process above, yt t = 1 : 500 using parameters = 09 vx = vy = 1(c) Draw the
Let yt = 1yt 1 + 2yt 2+ t with t N(01) Plot the spectra of yt in the following cases:(a) When the AR(2) characteristic polynomial has two real reciprocal roots given by r1 = 09 and r2 = 095(b) When the AR(2) characteristic polynomial has a pair of complex re ciprocal roots with modulus r = 095 and
Show that the spectral densities of AR(1) and MA(1) processes are given by (3.13) and (3.14), respectively.
Consider the three time series of concentrations of luteinizing hormone in blood samples from Diggle (1990). Perform a Bayesian spectral analysis of such series based on the single-harmonic regression model presented in Section 3.1.1.
Consider the Southern Oscillation Index series (soi.dat) shown in Fig ure 1.7 (a). Perform a Bayesian spectral analysis of such series.
Consider the UK gas consumption series ukgasconsumption.dat ana lyzed in Example 3.3. Analyze these series with models that include only the signi cant harmonics for p = 12. Compare the tted values obtained from such analyses to those obtained with the model used in Example 3.3.
Show that the reference analysis of the model in (3.2) leads to the ex pressions of p( v y) p(y) and p( y) given in Section 3.1.1.
Suppose yt follows a stationary AR(1) process with AR parameter and innovation variance v with ( v) uncertain. At any time t write Dt for the past data and information, including all past observations.If no additional information arises over the time interval (t 1 t] then Dt sequentially updates as
Twounivariatetimeseriesyt zt followthecoupleddynamicmodelsover t=1 givenby yt= yt 1+ zt+ t and zt= zt 1+ t where t N(0v) and t N(0u) are independent andmutually independent innovationssequences. Inbackshiftoperatornotation,(B)yt= zt+ t and (B)zt= t where (B)=1 Band (B)=1 BThetwoAR(1)coe cients
Thisquestionconcernsthealternativestate-spacerepresentationofan AR(p)model thatarisesasaspecialcaseofthestate-spacerepresenta tionofARMA(pq)modelswhenq=0.This isalsoeasilyseentobe de nedbyadirectlineartransformationofstatevectorsinthestandard
Consider the detrended oxygen isotope data analyzed in Chapter 5 (see also Aguilar, Huerta, Prado, and West 1999).(a) Under AR models use the AIC/BIC criteria to obtain the model order p that is the most compatible with the data.(b) Fit an AR model with the value of p obtained in (a) via maximum
Consider an ARMA(11) process with AR parameter MA parameter and variance v.(a) Simulate 400 observations from a process with = 09 = 06 and v =1(b) Compute the conditional least squares estimates of and based on the 400 observations simulated above.(c) Implement a MCMC algorithm to obtain samples
For the EEG data discussed in Section 2.3, perform the AR(8) Bayesian reference analysis as described there.(a) Draw histograms of the marginal posterior distributions of the model coe cients j for j = 1 : 8(b) Draw histograms of the marginal posterior distributions of the moduli and wavelengths of
Consider the quarterly US macro-economic data in Figure 2.15. Let yt be the implied series of quarterly changes (i.e., di erence values of quarterly actual in ation levels). Fit the reference analysis of an AR(8) model to the yt data and address the following.(a) If the AR(8) is accepted as a good
Sample 500 observations from a stationary AR(4) process with two com plex pairs of conjugate reciprocal roots. More speci cally, assume that one of the complex pairs has modulus r1 = 09 and frequency 1 = 5 while the other has modulus r2 = 075 and frequency 2 = 135Graph the simulated series and
Figure 2.14 plots the monthly changes in the US S&P stock market index over 1965 to 2016. Consider an AR(1) model as a very simple exploratory model for understanding local dependencies but not for forecasting more than a month or two ahead. We know there is a great deal of variation across the
Consider a process of the form yt = 2t+ t+05 t 1 iid tN(0v)(a) Find the ACF of this process.(b) Now de ne zt = yt yt 1+2What kind of process is this? Find its ACF.
Consider the AR(2) process yt = 1yt 1 + 2yt 2 + t with t N(0v)independent with 1 = 09 and 2 = 09 Is this process stable? If so write the process as an innite order MA process, yt = j=0 j t j Find j for all j
Consider the innite order MA process de ned by yt = t+a( t 1+ t 2+ )where a is a constant and the ts are i.i.d. N(0v) random variables.(a) Show that yt is nonstationary.(b) Consider the series of rst di erences zt = yt yt 1 Show that zt is a MA(1) process. What is the MA coe cient of this
Let xt be an AR(p) process with characteristic polynomial x(u) and yt be an AR(q) process with characteristic polynomial y(u)What is the structure of the process zt with zt = xt + yt?
Consider a MA(2) process.(a) Find its ACF.(b) Use the innovations algorithm to obtain the one-step-ahead predictor and its mean square error.
Consider the ARMA(1,1) model described by yt = 095yt 1 +08 t 1+ t with t N(01) for all t(a) Show that the one-step-ahead truncated forecast is given by yt t+1 =095yt + 08 t twith t tcomputed recursively via t j095yj 1 08 t j 1 , for j = 1 : t with t 0 = 0 and y0 = 0.(b) Show that the approximate
You observe yt = xt + t = 12 where xt follows a stationary AR(1) process with AR parameter and innovation variance v, i.e., xt = xt 1 + t with independent innovations t N(0v) Assume all parameters (v) are known.(a) Identify the ACF and PACF of yt, and comment of comparisons with those of xt(b) What
Suppose you observe yt = xt + t where:xt follows a stationary AR(1) process with AR parameter and inno vation variance v, i.e., xt = xt 1 + t with independent innovations N(0v);The t are independent measurement errors with t N(0w);The t and t series are mutually independent.It easily follows that q
Consider the AR(1) model given by(1 where t N(0v)where t N(0v)B)(yt) = t(a) Find the MLEs for and when =0 (b) Assume that v is known, = 0 and that the prior distribution for is U( 01) Find an expression for the posterior distribution of
Show that Equations (2.38) and (2.39) hold by taking expected values in (2.36) and (2.37) with respect to the whole past history y t.
Find the ACF of a general ARMA(1,1) process.
Verify the ACF of a MA(q) process given in (2.33).
Show that a prior on the vector of AR(p) coe cients of the form N( 10w 1) and N( j j 1w j) for 1 < j p can be written as p( ) = N( 0A1w) where A = H H with H and de ned in Section 2.4.2.
Verify that the expressions for the conditional posterior distributions in Section 2.4.1 are correct.
Plot the corresponding forecast functions for the AR(2) processes con sidered in Example 2.1.
Show that if an AR(2) process has a pair of complex roots given by r exp( i ) they can be written in terms of the AR coe cients as r =2 and cos( ) = 1 2r.
Show that, when the characteristic roots are all di erent, the forecast function of an AR(p) process has the representation given in (2.8).
Show that the general solution of the homogeneous di erence Equation(2.9) has the form (2.10).
Consider the AR(2) series yt = 1yt 1 + 2yt 2 + t with t N(0v).Following Section 2.1.2, rewrite the model in the standard DLM form yt = Fxt and xt =Gxt 1+Ft where F= 1 0xt = yt yt 1 G= 1 2 10 We know that this implies that, for any given t and over k 0 the forecast function is E(yt+kxt) = FGkxt(a)
Show that the eigenvalues of the matrix G given by (2.7) correspond to the reciprocal roots of the AR(p) characteristic polynomial.
This question concerns a time series model for continuous and positive outcomes yt. Suppose a series xt follows a stationary AR(1) model with parameters v and the usual normal innovations. De ne a transformed time series yt = exp( +xt) for each t for some known constant(a) Show that yt a rst-order
Consider an AR(2) process with AR coe cients = ( 1 2).(a) Show that the process is stationary for parameter values lying in the region 1 < 2
Suppose yt follows a stationaryAR(1)model withARparameter and innovationvariance v. De ne x= (y1 yn).We knowthat x N(0s n)wheres=v (1 2) is themarginal varianceof the ytprocessandthecorrelationmatrix nhas(i j)element i j viz.n=1 2 n 1 1 n 2 2 1 n 3 n 1 n 2 n 3 1 FindtheprecisionmatrixKn=s 1 1 n
Consider theAR(1)processyt= yt 1+ t with t N(0v) Show thattheprocessisnonstationarywhen = 1
ConsidertheAR(1)processyt= yt 1+ t with t N(0v) If
Sample T = 1000 observations from model (1.1) with d = 1 using your preferred choice of values for(i) and vi for i = 12 with (i) < 1 and (1)= (2) Assuming that d is known and using prior distributions of the form p( (i)) = N(0c), for i = 12 and some c > 0, p( ) =U(aa) and p(v) = IG( 0 0), with a >
Refer to the conjugate analysis of the AR(1) model in Example 1.7.Using the fact that ( yFv) N(mvC), nd the posterior mode of v via the EM algorithm.
Consider the following models:yt = 1yt 1+ 2yt 2+ t yt = acos(2 0t)+bsin(2 0t)+ t(1.26)(1.27)with t N(0v) and 0 >0 xed.(a) Sample T = 200 observations from each model using your favorite choice of the parameters. Make sure your choice of ( 1 2) in model(1.26) lies in the stationary region. That is,
Show that the distributions of ( yF) and (vyF) obtained for the AR(1) conjugate analysis using the conditional likelihood are those given in Example 1.7.
Show that the expressions for the distributions of (yFv) (vyF), and( yF)under the conjugate analysis in Section 1.5.3 are those given on page 24.

Showing 1 - 100 of 888

Time Series Modeling Computation And Inference 2nd Edition Raquel Prado, Marco A. R. Ferreira, Mike West - Solutions

Step by Step Answers