## Lecture Notes on Least Squares Prediction Theory

In these notes we establish some basic results for least squares prediction theory. These results are useful in a variety of contexts. For instance, they are valuable for solving linear rational expectations models, representing covariance stationary time series processes, and obtaining martingale difference decompositions of strictly stationary processes.

The basic mathematical construct used in these notes is an inner product defined between two random variables. This inner product is calculated by taking the expectation of the product of the two random variables. Many of the results obtained using this particular inner prod- uct are analogous to results obtained using the standard inner product on multi-dimensional Euclidean spaces. Hence intuition obtained for Euclidean spaces can be quite valuable in this context as well.

The formal mathematical machinery that is exploited in these notes is the Hilbert space theory. There is a variety of references on Hilbert spaces that should provide good complementary reading, e.g. Hal- mos (1957) and Luenberger (1969).

**Pages:**13-44|

**Title of book:**Rational Expectations Econometrics|

**Editor(s):**Lars Peter Hansen and Thomas J. Sargent|

**Place of Publication:**Boulder and Oxford|

**Publisher:**Westview Press|

**Tags:**Econometrics|Export BibTeX >

@article{hansen:1991lecture, title={Lecture Notes on Least Squares Prediction Theory}, author={Hansen, LP and Sargent, TJ}, journal={Rational Expectations Econometrics}, year={1991} }✕

## Exact Linear Rational Expectations Models: Specification and Estimation

A distinguishing characteristic of econometric models that incorporate rational expectations is the presence of restrictions across the parameters of different equations. These restrictions emerge because people’s decisions are supposed to depend on the stochastic environment which they confront. Consequently, equations describing variables af- fected by people’s decisions inherit parameters from the equations that describe the environment. As it turns out, even for models that are linear in the variables, these cross-equation restrictions on the parameters can be complicated and often highly nonlinear.

**Pages:**45-76|

**Title of book:**Rational Expectations Econometrics|

**Editor(s):**Lars Peter Hansen and Thomas J. Sargent|

**Place of Publication:**Boulder and Oxford|

**Publisher:**Westview Press|

**Tags:**Econometrics|Export BibTeX >

@techreport{hansen1981exact, title={Exact Linear Rational Expectations Models: Specification and Estimation}, author={Hansen, Lars Peter and Sargent, Thomas J and others}, year={1981}, institution={Federal Reserve Bank of Minneapolis} }✕

## Two Difficulties in Interpreting Vector Autoregressions

The equilibrium of a typical dynamic rational expectations model, is a covariance stationary (n x 1) vector stochastic process z(t). This stochastic process determines the manner in which random shocks to the environment impinge over time on -agents’ decisions and ultimately upon market prices and quantities. Surprises, Le., random shocks to agents’ information sets, prompt revisions in their contingency plans, thereby impinging on equilibrium prices and quantities.

**Pages:**77-119|

**Title of book:**Rational Expectations Econometrics|

**Editor(s):**Lars Peter Hansen and Thomas J. Sargent|

**Place of Publication:**Boulder and Oxford|

**Publisher:**Westview Press|

**Tags:**Econometrics|Export BibTeX >

@article{hansen:1991, title={Two Difficulties in Interpreting Vector Autoregressions}, author={Hansen, Lars Peter and Sargent, Thomas J}, journal={Rational Expectations Econometrics}, volume={1}, pages={77--119}, year={1991}, publisher={Westview Press Boulder, CO} }✕

## Time Series Implications of Present Value Budget Balance and of Martingale Models of Consumption and Taxes

**Pages:**121-161|

**Title of book:**Rational Expectations Econometrics|

**Editor(s):**Lars Peter Hansen and Thomas J. Sargent|

**Place of Publication:**Boulder and Oxford|

**Publisher:**Westview Press|

**Tags:**Econometrics, Financial Market Linkages to the Macroeconomy|Export BibTeX >

@article{hansen:1991, title={Time Series Implications of Present Value Budget Balance and of Martingale Models of Consumption and Taxes}, author={Hansen, Lars Peter and Roberds, William and Sargent, Thomas J}, journal={Rational Expectations Econometrics}, pages={121--61}, year={1991}, publisher={Westview Press Cambridge} }✕

## Faster Methods for Solving Continuous Time Recursive Linear Models of Dynamic Economies

**Pages:**177-208|

**Title of book:**Rational Expectations Econometrics|

**Editor(s):**Lars Peter Hansen and Thomas J. Sargent|

**Place of Publication:**Boulder and Oxford|

**Publisher:**Westview Publisher|

**Tags:**Econometrics|Export BibTeX >

@article{hansen:1991, title={Faster Methods for Solving Continuous Time Recursive Linear Models of Dynamic Economies'}, author={Hansen, Lars Peter and Heaton, John and Sargent, Thomas J}, journal={Rational Expectations Econometrics}, pages={177--208}, year={1991} }✕

## Prediction Formulas for Continuous Time Linear Rational Expectations Models

In this note we derive optimal prediction formulas to be used in solving continuous time rational expectations models. In these deriva- tions we employ Laplace transforms in a manner analogous to the use of z transforms for solving discrete time optimal prediction problems in Hansen and Sargent (1980a, Appendix A). The formulas are intended to play the same role for continuous time models that the discrete time formulas for optimal predictions of-geometric distributed leads did in Hansen and Sargent (1980a).

**Title of book:**Rational Expectations Econometrics|

**Editor(s):**Lars Peter Hansen and Thomas J. Sargent|

**Place of Publication:**Boulder and Oxford|

**Publisher:**Westview Press|

**Tags:**Econometrics|Export BibTeX >

@article{hansen:1991,

title={Prediction Formulas for Continuous Time Linear Rational Expectations Models}, author={Hansen, Lars Peter and Roberds, William and Sargent, Thomas J}, journal={Rational Expectations Econometrics}, year={1991}, publisher={Westview Press Cambridge} }✕

## Identification of Continuous Time Rational Expectations Models from Discrete Time Data

This paper proves two propositions about identification in a continuous time version of a linear stochastic rational expectations model. The model is a continuous time version of Lucas and Prescott (1971), in which’the equilibrium can be interpreted.~ the solution of a ‘stochastic control problem, either of a collection of private agents or of a fictitious “socialplanner. Estimation is directed toward isolating the parameters of the agent’s objective function and of the stochastic processes of the forcing functions that the agent faces. This approach has been advocated by Lucas (1967, 1976), Lucas and Prescott (1971), and Lucas and Sargent (1981) as offering the potential to analyze an interesting class of policy interventions promised by structural models, while meeting the criticisms of most econometric policy evaluation methods that were made by Lucas (1976). At the same time, inspired by the work of Sims (1971), Geweke (1978), and P.C.B. Phillips (1972, 1973, 1974), we want to estimate models in which optimizing economic agents make decisions at finer time intervals than the interval of time between the observations used by the econometrician. We adopt a continuous time theoretical framework both because it is an interesting limiting case because it has received extensive attention in the theoretical and the econometric literatures.

**Pages:**209-218|

**Title of book:**Rational Expectations Econometrics|

**Editor(s):**Lars Peter Hansen and Thomas J. Sargent|

**Place of Publication:**Boulder and Oxford|

**Publisher:**Westview Press|

**Tags:**Econometrics|Export BibTeX >

@article{sargenthansen:1991,

title={Identification of Continuous Time Rational Expectations Models From Discrete Time Data},

author={Sargent, Thomas J. and Hansen, Lars P.},

journal={Rational Expectations Econometrics},

year={1991}

}

✕## Using Conditional Moments of Asset Payoffs to Infer Volatility of Intertemporal Marginal Rates of Substitution

Previously Hansen and Jagannathan (1990a) derived and computed mean-standard deviation frontiers for intertemporal marginal rates of substitution (IMRS) implied by asset market data. These frontiers give the lower bounds on the standard deviations as a function of the mean. In this paper we develop a strategy for utilizing conditioning information efficiently, and hence improve on the standard deviation bounds computed by Hansen and Jagannathan. We implement this strategy empirically by using the seminonparametric (SNP) methodology suggested by Gallant and Tauchen (1989) to estimate the conditional distribution of a vector of monthly asset payoffs. We use the fitted conditional distributions to calculate both conditional and unconditional standard deviation bounds for the IMRS. The unconditional bounds are as sharp as possible subject to robustness considerations. We also use the fitted distributions to compute the moments of various candidate marginal rates of substitution suggested by economic theory, and in particular the time-nonseparable preferences of Dunn and Singleton (1986) and Eichenbaum and Hansen (1990). For these preferences, our findings suggest that habit persistence will put the moments of the IMRS inside the frontier at reasonable values of the curvature parameter. At the same time we uncover evidence that the implied IMRS fails to satisfy all of the restrictions inherent in the Euler equation. The findings help explain why Euler equation estimation methods typically find evidence in favor of local durability instead of habit persistence for monthly data.

**Journal:**Journal of Econometrics|

**Volume:**45|

**Pages:**141-179|

**Tags:**Econometrics, Financial Market Linkages to the Macroeconomy|Export BibTeX >

@article{gallant1990using, title={Using Conditional Moments of Asset Payoffs to Infer the Volatility of Intertemporal Marginal Rates of Substitution}, author={Gallant, A Ronald and Hansen, Lars Peter and Tauchen, George}, journal={Journal of Econometrics}, volume={45}, number={1-2}, pages={141--179}, year={1990}, publisher={Elsevier} }✕

## Estimating Models with Intertemporal Substitution Using Aggregate Time-Series Data

In conducting empirical investigations of the permanent income model of consumption and the consumption-based intertemporal asset pricing model, various authors have imposed restrictions on the nature of the substitutability of consumption across goods and over time. In this paper we suggest a method for testing some of these restrictions and present empirical results using this approach. Our empirical analyses focuses on three questions: (i) Can the services from durable and nondurable goods be treated as perfect substitutes? (ii) Are preferences completely separable between durable and nondurable goods? (iii) What is the nature of intertemporal substitutability of nondurable consumption? When consumers’ preferences are assumed to be quadratic, there is very little evidence against the hypothesis that the services from durable goods and nondurable goods are perfect substitutes. These results call into question the practice of testing quadratic models of aggregate consumption using data on nondurables and services only. When we consider S branch specifications, we find more evidence against perfect substitutability between service flows, but less evidence against strict separability across durable and nondurable consumption goods. Among other things, these findings suggest that the empirical shortcomings of the intertemporal asset pricing model cannot be attributed to the neglect of durable goods.

**Journal:**Journal of Business & Economic Statistics|

**Volume:**8|

**Issue Number:**1|

**Pages:**53-69|

**Tags:**Econometrics|Export BibTeX >

@article{eichenbaum1990estimating, title={Estimating Models with Intertemporal Substitution Using Aggregate Time Series Data}, author={Eichenbaum, Martin and Hansen, Lars Peter}, journal={Journal of Business & Economic Statistics}, volume={8}, number={1}, pages={53--69}, year={1990}, publisher={Taylor & Francis} }✕

## Efficiency Bounds Implied by Multiperiod Conditional Moment Restrictions

In this article we study a class of econometric models that imply a set of multiperiod conditional moment restrictions. These restrictions depend on an unknown parameter vector. We construct an extensive class of consistent, asymptotically normal estimators of this parameter vector and calculate the greatest lower bound for the asymptotic covariance matrices of estimators in this class. In so doing, we extend results reported by Hansen (1985) and Stoica, Soderstrom, and Friedlander (1985), by allowing for more general forms of nonlinearities and temporal dependence. Many dynamic econometric models imply that the expectation of a function of a currently observed data vector and an unknown parameter vector conditioned on information available at some point in the past is 0. We focus on models in which the conditioning information is lagged more than one time period, as in the models considered by Barro (1981), Dunn and Singleton (1986), Eichenbaum and Hansen (1987), Eichenbaum, Hansen, and Singleton (1988), Hansen and Hodrick (1983), Hansen and Singleton (1988), and Hall (1988). Hence we consider econometric models that imply multiperiod conditional moment restrictions that depend on an unknown parameter vector. Within the context of these models, it is possible to estimate the parameter vector without simultaneously estimating the law of motion for the entire set of observable variables. The basic idea is to use the conditional moment restrictions to deduce a set of unconditional moment restrictions. Then estimators of the parameter vector can be obtained by using sample counterparts to the unconditional moment restrictions as described by Sargan (1958) and Hansen (1982). Such estimators are referred to as generalized method of moments (GMM) estimators. For most applications the conditional moment restrictions imply an extensive set of unconditional moment restrictions. As a consequence, there is a vast array of GMM estimators that can be used to estimate consistently the parameter vector of interest. Each member of this set of estimators is constructed using a distinct collection of the unconditional moment restrictions. Hence it is of interest to compare the performances of the alternative GMM estimators. For tractability we investigate only the asymptotic distributions of the estimators in question. More precisely, we use a method suggested by Hansen (1985) for calculating a greatest lower bound for the asymptotic covariance matrices of the alternative GMM estimators, that is, an efficiency bound. We compute the efficiency bound for a rich collection of time series models that imply multiperiod conditional moment restrictions. Hansen (1985) illustrated this method for a time series model with conditionally homoscedastic moving-average disturbance terms for which the moving-average polynomial is invertible. Stoica et al. (1985) calculated efficiency bounds for GMM estimators for autoregressive parameters in autoregressive moving-average models without unit roots. They established that the efficiency bound for GMM estimators of the autoregressive parameters coincides with the asymptotic covariance matrix of the Gaussian maximum likelihood estimators. The models considered by Hansen (1985) in his illustrative example and by Stoica et al. (1885) can be viewed as special cases of the models considered in this article. Although we do not make any direct comparisons to maximum likelihood, we do allow for moving-average disturbances that are conditionally heteroscedastic and moving-average lag polynomials that cannot be inverted.

**Journal:**Journal of the American Statistical Association|

**Volume:**83|

**Issue Number:**403|

**Pages:**863-871|

**Tags:**Econometrics|Export BibTeX >

@article{hansen1988efficiency, title={Efficiency Bounds Implied by Multiperiod Conditional Moment Restrictions}, author={Hansen, Lars Peter and Heaton, John C and Ogaki, Masao}, journal={Journal of the American Statistical Association}, volume={83}, number={403}, pages={863--871}, year={1988}, publisher={Taylor & Francis Group} }✕