Examples

Examples

1. Since log $(n)/n\to 0$ then $\log (n)=o(n)$

2. For k fixed, $\left(\genfrac{}{}{0ex}{}{n}{k}\right)=o(n^k)$ . To see this, notice that

$$\begin{array}{c}(\begin{array}{l}n\\ k\end{array})n^{-k}=\frac{n(n-1)\cdots (n-k+1)(n-k)\cdots 1}{n^{k}k(k-1)\cdots 1(n-k)(n-k-1)\cdots 1}\\ =\frac{n(n-1)\cdots (n-k+1)}{n^{k}k(k-1)\cdots 1}\\ <\frac{n(n-1)\cdots (n-k+1)}{n^k}\\ 0.\end{array}$$

We pronounce $o\left(a_n\right)$ as “little oh of order ${a_n}^{\mathfrak{n}}$ and $O(a_n)$ as “big oh of order ${\mathbf{a}}_{\mathbf{n}}{.}^{\ast }$ The little oh terminology is shorthand for “disappears at this rate and any faster ${\mathrm{rate}}^{\mathfrak{n}}$ ; the big oh terminology means that for division by the specified rate the variable “hangs around,” i.e., it does not get too large or too small.

Extensions to random sequences, $x_n,$ , follow naturally, although they give rise to a more complex idea of “convergence”.

Definition. For a random sequence $\{X_n{\}}_{n=1}^{\infty }$ and deterministic sequence $\{a_n{\}}_{n=1}^{\infty }$ , we say that $\begin{array}{r}{X}_n=o_p(a_n)ifp{\lim }_{n\to \infty }(X_n/a_n)=0\end{array}$ . We say that $X_n$ is $O_P(a_n)$ if there exists a constant K such that $P(|x_n/a_n|<K)>1-\epsilon$ for all $\epsilon >0$ and all n sufficiently large.

As an example, suppose $X_1,\dots ,X_n\sim$ ind N(0, 1), so $\bar{X}\sim \mathrm{N}(0,n^{-1})$ . Now $P[|\bar{X}|>\epsilon ]0$ as $n\to \infty$ by Chebyshev, so $\bar{X}$ is $o_p(1)$ . Further, consider $Z_n=$ $n^{1/2}\bar{X}$ . By the central limit theorem, $Z_n{\to }^d\mathrm{N}(0,1)$ . Let $\mathcal{N}(t)$ be the c.d.f. of a standard normal variable so $P(n^{1/2}{\bar{X}}_n<K)\to \mathcal{N}(K)$ . Choose K such that $P(|n^{1/2}{\bar{X}}_n|<K)\to \mathcal{N}(K)-\mathcal{N}(-K)>1-\epsilon ,\mathrm{so}{\bar{X}}_n\mathrm{is}{O}_p(n^{-1/2})$

We pronounce this convergence as “little oh p of order $a_n{.}^{\mathfrak{N}}$ We typically use such convergence as shorthand, i.e., rather than expressing completely a term that disappears, we summarize it with a term that indicates the order at which it vanishes.

In chapters 17 and 20, we employ limit distributions of objects of interest that are expressed as functions of Brownian motions rather than through the usual asymptotic normal distribution. Here we give a short overview of the area, omitting technical details but providing pointers to the literature.

Just as sums (or averages) of suitably normalized random variables ${\sum }_{i=1}^n{x}_i$ are often well approximated in the limit by a normal distribution via the central limit theorem, suitably normalized partial sums ${\sum }_{i=1}^{\left[ns\right]}{x}_i$ (where $s\in (0,1]$ and [·] takes the integer portion of its argument) are often well approximated by convergence to a Brownian motion using functional central limit theory (FCLT). In such cases, continuous functions of normalized sums will converge to that function of a normal random variable and—via the continuous mapping theorem—continuous functions of normalized partial sums converge to that function of a Brownian motion. The extension to the use of functional central limit theory allows a wider set of results to be established and is the natural tool for obtaining asymptotic distributional approximations in situations where partial sums arise.

Brownian motions are distributions for paths on [0, 1] that satisfy some basic requirements. Consider $W(s)$ for $s\in [0,1]$ . If (a) W(0) = 0, (b) $W(b)-W(a)\sim$ $\mathrm{N}(0,b-a)$ for $b>a$ and $(\mathsf{c})W(\mathbf{c})-W(\mathbf{b})$ is independent of $W(b)-W(a)$ for $0\le a<b<c\le 1$ , then $W(s)$ is called a standard Brownian motion on $s\in [0,1]$ Using (b) with $a=0,b=1$ , this implies that $W(1)=\mathrm{N}(0,1)$ , making standard Brownian motion an extension of the standard normal distribution.

Convergence of partial sums to Brownian motions via the FCLT can be written $n^{-1/2}{\sum }_{i=1}^{[ns]}{x}_i\Rightarrow \overline{W(s)}$ , where ⇒ denotes weak convergence. Results require assumptions on the sequence $\{x_i{\}}_{i=1}^n$ , and as with central limit distributions there exist a wide range of assumptions that result in convergence; a useful textbook reference is White (2001, chapter 7). A thorough examination is also provided in Davidson (1994).

The FCLT can be used to analyze the process $y_t=\rho y_{t-1}+u_t$ , where $\rho =1-$ $\gamma /T$ . Let ${\omega }^2$ equal the spectral density of $u_t$ at frequency 0, which is nonnegative and assumed to be finite. If $\rho =1(\gamma =0)$ and $T^{-1/2}{\bar{u}}_1\to \dot{{}^p}0$ , we have $T^{-1/2}{\omega }^{-1/2}{y}_t=$ $T^{-1/2}{\omega }^{-1/2}{\sum }_{i=1}^{[Ts]}{u}_t\Rightarrow \stackrel{\cdot }{W(s)}$ under suitable assumptions on the shocks $u_t.\mathrm{If}\gamma >0.$ the FCLT can still be applied after some rearrangement and, under the same assumptions, $\begin{array}{r}{T}^{-1/2}{\omega }^{-1/2}\widehat{y_{[Ts]}}\Rightarrow {\int }_0^s{e}^{-\gamma (s-r)}dW(r)\end{array}$ via a continuous mapping theorem (Bobkoski, 1983; Phillips, 1987). This process is known as a standard Ornstein– Uhlenbeck process.

While some functions of standard Brownian motion have distributions that are normal, such as the endpoint in the previous paragraph, other cases such as the function $\int W(s{)}^{2}ds$ are not normally distributed. In practice, many interesting functions map the Brownian motion to $\mathbb{R},$ and hence the resulting distribution can be characterized through its mean, variance, quantiles, and density even though it is typically not normal. Because the distributions are often not standard, well-known, and tabulated distributions, statistics that have these distributions as asymptotic limits require the provision of tables of critical values. See Tanaka (1996) for details on computing critical values.

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Abadir, K. M.

absolute error loss, 20. See also mean absolute error (MAE) loss

Aggarwal, R.

Aiolfi, M., 319, 320, 462–63

Akaike, H., 96, 120

Albert, A., 264

Al-Osh, M.

Alzaid, A. A.

Amato, J. D.

Amemiya, T., 270

American Association of Individual Investors, 322

Amihud, Y.

Amisano, G., 338, 425–26

Andersen, T.G., 284, 284, 285, 289

Andreou, E.

Andrews, D. W., 384, 449, 472n3, 483

Andrews’ SupF test, 450

Andrews–Ploberger (AP) test, 450

Ang, A., 179, 320n5, 445

ARIMA models, 39, 154, 155, 163, 391; success of, 165; univariate ARIMA models, 164

artificial neural networks (ANN), 249–50, 332; and White’s QuickNet Algorithm, 251–52, 251n5

Artis, M., 23

Aruoba, S. B., 224, 502, 503

asset allocation with return predictability, 203–6
Athanasopoulos, G., 187n1

autoregressive (AR) models, 134, 137, 138, 139–40, 150, 152, 152n6, 156, 163, 175, 183, 184, 212–13, 223, 241, 303, 332, 427; AR(1) model, 468, 474, 478, 479, 480, 483, 486, 492, 499, 507, 510, 511; AR(2) model, 506, 511; AR(4) model, 235, 480; AR(BIC) method, 185; AR forecasts, 239; AR( p) regression model, 153–54; and empirical evidence, 170–72; exponential smooth transition autoregressive (ESTAR) models, 169; performance of, 428; smooth transition autoregressive (STAR) models, 169–71, 172–73, 180, 183, 184, 332–33; threshold autoregressive (TAR) models, 167–69, 170, 172–73, 180; with lag length selected by AIC, 427, 480. See also autoregressive moving average (ARMA) models, forecasting with; vector autoregressions (VARs)

autoregressive conditional duration (ACD), 494, 495

autoregressive conditional heteroscedasticity (ARCH) models, 280, 281, 308, 367; the ARCH(q) model, 280; multivariate ARCH models, 307

autoregressive moving average (ARMA) models, 7, 11, 99, 133–34, 137–42, 165, 226, 243, 494; as approximations, 134; AR representation of the MA(1) process, 138–40, 139n2; the ARMA-GARCH model, 67; as the backbone of commercial forecasts, 133; covariance stationarity of, 134–37; forecast errors for the AR(1) model, 138; linear ARMA models, 7; minimal information requirements of, 133; popularity and success of, 133; resiliency of in empirical work, 134. See also autoregressive moving average (ARMA) models, deterministic and seasonal components of; autoregressive moving average (ARMA) models, estimation and lag selection for; autoregressive moving average (ARMA) models, forecasting with

autoregressive moving average (ARMA) models, deterministic and seasonal components of, 155; deterministic time trends, 158; forecasting of models with seasonal components, 155–57. See also Holt-Winters procedure

autoregressive moving average (ARMA) models, estimation and lag selection for, 142; choice of lag orders, 144–46; and state-space representations of ARMA models, 143–44, 144n3

autoregressive moving average (ARMA) models, forecasting with, 147; classical theory of, 147–50; direct versus iterated multiperiod forecasts, 153–54, 154n9; finite-sample properties of forecasts from AR(1) models, 151–53, 151nn5–6, 207; and forecasting variables with unit roots, 154–55; forecasts generated by the AR(1) model, 147–48; forecasts generated by the MA(1) model, 148–49; MSE (mean squared error) for the AR(1) process, 150. See also Wiener-Kolmogorov prediction formulas

Bagging, 90, 104–5, 107, 108, 113, 122, 235

Bai, J., 72, 219, 227n5, 230, 231, 240, 259, 436, 436n4, 449, 451, 451n5, 454, 455, 456

Baillie, R. T., 242, 285

Baltagi, B. H., 242

Banbura, M., 200, 201, 202, 236, 499, 500, 500n5, 501

bandwidth, 245, 246, 253, 256, 292

Bank of England, 308

Barberis, N., 59, 60, 85, 205

Barndorff-Nielsen, O. E., 289

Basic StepM method, 415

Basset, G., 303

Batchelor, R., 355

Bates, J. M., 317, 320, 326, 330

Bauer, M. D., 190

Bauwens, L., 457, 494, 495

Bayesian forecasting Bayesian forecasting

models/methods/approaches, 4, 6, 40, 54–56, 76, 88, 188, 235; application of to stock returns, 342–43; Bayes risk, 54–55, 77–80; Bayesian investor’s asset allocation (example of Bayesian decision making), 86–88; Bayesian linear regression models, 229; Bayesian model averaging (BMA), 9, 339–41, 342–43; Bayesian modeling of the predictive distribution, 80–81; Bayesian posterior density estimation problem, 77; Bayesian prediction problems, 22; comparing Bayesian and classical methods, 56–57; comparing Bayesian and classical methods (example of [asset allocation with parameter uncertainty]), 59–62; computational/numerical methods of, 83–85; construction of a Bayesian decision rule, 54; construction of prior distributions in Bayesian analysis, 15; difficulties of estimating and forecasting with VARs, 7; economic applications of, 85–86; empirical Bayes methods, 80. See also Bayesian VARs (BVARs); density forecasts, for binary outcomes: Bayesian approaches to; information criteria (IC): Bayes/Bayesian information criterion (BIC)

Bayesian VARs (BVARs), 194–95, 212, 213, 214; and alternative priors for, 198–99; Bayesian estimation of, 195–96; empirical example of (asset allocation with return predictability), 203–6; empirical performance, 202–3; factor models versus Bayesian VARs, 236; large-dimensional VARs, 200–202; and the Minnesota priors, 196–97, 212, 213; and time-varying parameter (TVP) VARs, 199–200

Bekaert, G., 179, 320n5, 445

Belloni, A., 104, 106, 106n9

Bennett, P., 13

Berkowitz, J., 437

Bernanke, B. S., 194

Bernoulli distribution, 261, 265, 441, 448

Bernoulli random variable, 260n1, 448, 492, 492n2

bilinear process, 141–42 binary forecasts, 260–61, 273–74; the binary forecasting problem, 268, 423, 425; forecasts of binary variables, 8, 69–70; point and probability forecasts for binary outcomes, 261–64, 262n2, 263n3. See also binary outcomes, construction of point forecasts for; density forecasts, for binary outcomes

binary outcomes, construction of point forecasts for, 269; comparison of estimation models, 271–72; forecasts via p(Z), 269; using discriminant analysis for, 270, 270n8; using maximum utility extension for, 270–71

Blue Chip survey/forecasts, 209, 322, 369

Boivin, J., 194, 225, 235

Bollerslev, T., 280, 282, 284, 285, 289, 305

bonds: bond prices, 179; government bonds, 202; zero-coupon bonds, 188

Bonferroni bound, 413, 415, 483

boosted regression trees, 245, 256–59

“boundedness,” 12, 16

Box, G. E., 7, 133, 144, 155

Boyes, W. J., 25

break model example, 137

break process, forecasts that model the, 456; change point models, 457; Markov switching models, 456; time-varying parameter (TVP) random walk models, 458–59

breaks, ad hoc methods for dealing with, 460; and forecast combination, 462–63; intercept correction, 461–62; weighting schemes that downweight past observations, 460

Breiman, L., 72, 104

Breitung, J., 219

Brier, G. W., 424

Brier Quadratic Probability Score (QPS), 30, 264, 424–25, 429

Brockwell, P. J., 134

Brown, R., 161

Brown, R. L., 452

Brownian bridge, 437

Brownian motion, 289, 405, 436, 468, 468n1, 472, 482; and functional central limit theory, 515

Burman, P., 99

Busetti, F., 400

Calibration, 422, 432, 441; of distributional forecasts, 428, 429–31; squared calibration errors, 354, 428–29

Campbell, J. Y., 186, 193, 484

Canjels, E., 472

Capistrán, C., 22, 24, 323n7, 462–63

Cappiello, L., 306

Carriero, A., 202

Cavanagh, C. L., 483

CAVieR models, 303, 306

certainty equivalence, 33

certainty equivalent return (CER), 36, 112–13

Chatfield, C., 158

Chauvet, M., 71, 267

Chen, A.-S., 272

Chen, X., 74, 250

Chernozhukov, V., 104, 106, 106n9

Cheung, Y.-W., 489

Chib, S., 457

Chinn, M. D., 489

Cholesky decomposition, 305

Chong, Y. Y., 392, 394, 395

Chow, E., 99

Christoffersen, P. F., 22, 366, 367, 441, 479

Claeskens, G., 329

Clark, T. E., 200, 369, 392n1, 394, 395, 396, 397, 403, 404, 406, 407, 407n9, 408, 411n11, 420, 463

classical forecasting models, 47, 74; comparing Bayesian and classical methods, 56–57; loss-based versus two-step approaches, 48–50

Clemen, R. T., 337

Clements, M. P., 28, 175, 321, 461, 462

coefficient estimates, 84, 91, 94, 123, 151, 176, 226, 376, 378, 387, 392

combination weights, estimation of, 316; estimation methods for time-varying combination weights, 319–20; forecast of under MSE loss, 316–19, 317n3

combinations, 310–12, 311n1, 344, 462–63; application of to survey forecasts, 321–25; empirical comparison of forecast combinations, 341–42; the forecast combination puzzle, 320–21; risk for forecast combinations, 325–29, 327n8. See also combination weights, estimation of; optimal forecast combinations

complete subset regressions, 330–32

Confederation of British Industry, 322

Connor, G., 219

Consensus Economics, 322

constrained optimization methods, 73

continuous ranked probabilistic score (CRPS), 338

copulas, 306–8

Corradi, V., 426, 427, 436, 486

cost function, 30; assumption of symmetry for, 17

CPIAUCSL (Consumer Price Index for All Urban Consumers), 149

Cramer–von Mises (CvM) test, 435, 436–37, 465

cross validation, 90, 90n2, 99–101, 107, 108, 119, 160, 246, 250, 251–52, 251n5, 299, 378, 454; cross validation for sample mean, 100–101; the “leave-Tυ -out” cross validation approach, 99

Croushore, D., 496, 498

Dacco, R., 179

D’Agostino, A., 235

Dangl, T., 459

Daouk, H., 272

data: data mining, 413–15; data repositories, 490; data sets, 7; macroeconomic data, 170, 369, 496; mixed data sampling (MIDAS) methods, 501–2; real-time data, 495–96, 496n4, 498, 504; state-space approaches with irregular data, 502–4. See also data-generating process (DGP)

data-generating process (DGP), 3, 5, 355–56, 365, 366–67, 377, 378–79; forecasting models as simplified representations of evolving data-generating process, 11; shifts in the parameters of the data-generating process and model instability, 447

Davies, A., 389

Davies, R. B., 180

Davis, R. A., 134

Dawid, A. P., 8n2, 337

De Mol, C., 229

decision theory, 4, 17, 25, 40, 40n1

Del Negro, M., 206, 208, 209, 210

density combination, 312, 336–37, 339; classical approach to, 338

density forecasts, 8, 30, 57–59, 60n8, 185, 260, 308–9; comparison of Gaussian density forecasts, 430, 431, 432, 438; reporting of density forecasts, 300–301. See also density forecasts, approaches to; density forecasts, for binary outcomes; density forecasts, evaluation of; volatility, and density forecasting

density forecasts, approaches to, 291–92; nonparametric and semiparametric density estimation approaches, 297–300; semi-non-parametric (SNP) approach, 298–300. See also parametric density models

density forecasts, for binary outcomes, 265; Bayesian approaches to, 267–68; nonparametric approaches to, 268–69; and parametric density forecasting models, 265–67

density forecasts, evaluation of, 422; comparison of density forecasts, 425–27; and dynamic specification testing, 438n5; empirical application of to volatility forecasting, 427–28; evaluation based on loss functions, 423; evaluation of individual density forecasts, 423–25, 424n1; and tests based on the probability integral transform (PIT), 433–38, 433n2, 438

Dickey, D. A., 474

Diebold, F. X., 22, 110, 134, 157, 178, 224, 246n2, 311, 318, 358n4, 366, 391, 395, 398–99, 400, 416, 423, 435n3, 462, 473–74, 479, 502, 503. See also Diebold-Mariano (DM) test

Diebold-Mariano (DM) test, 110, 112, 184, 216, 239, 241, 333, 391, 392, 398–400, 409, 416, 420, 427–28, 465, 486; popularity of for comparing predictive accuracy across pairs of models, 420

Dimitras, A. I., 269, 270

distributional forecasts, 8, 428–29, 441–42; calibration of, 428, 429–31; and the comparison of Gaussian density forecasts, 430, 431, 432, 438; evaluation of, 11; and the Receiver Operator Characteristic (ROC) curve, 432–33; resolution of, 431–32; scoring rules for, 29–31; sharpness of, 432

Doan, T., 196

Dobrev, D., 236

Donsker’s theorem, 405

Dua, P., 355

Dueker, M., 268

Durbin, J., 85, 452

Durland, J. M., 178

dynamic specification, 89, 179; dynamic specification testing, 438n5

dynamic stochastic general equilibrium (DSGE) models, 85, 186, 206–8; empirical evidence of DSGE models, 209–10; estimation and computation of forecasts for DSGE models, 208–9

Eckmeier, S., 219

economic surveys, primary sources of, 322

economic variables, 5, 7, 134, 166, 218, 232, 236, 457; categories of, 235; distribution of, 157; variation across, 219

efficiency properties/tests: conventional forecast efficiency tests, 357n3; efficiency properties for the AR(1) process, 360; efficiency properties for a covariance stationary process, 359; efficiency properties under squared error loss, 358–63; efficiency properties with known loss, 355–58; interpreting efficiency tests, 368–71; and “test functions,” 357; “weak” versus “strong” forecast efficiency, 356, 356n2 versus “strong"forecast efficiency,356,356n2

Ehrbeck, T., 13

EKT (Elliott, Komunjer, Timmermann) loss functions, 23–24, 28–29

elastic net estimator, 72

electricity: consumption of, 157; electricity loads, 245; electricity prices, 179, 245

Eliasz, P., 194

Elliott, G., 16, 18, 23, 24, 25, 28, 36n15, 72, 262n2, 266, 271, 272, 314, 320, 330–32, 342, 364, 365, 371, 449, 450, 454, 472, 474, 476, 482, 483, 484. See also qLL test (of Elliott and Müller)

Embrechts, P., 307

employment, 28, 155, 202, 218, 235; employment costs, 207; forecasts of for New York State, 203

ENC-NEW test, 395, 397, 404, 408

ENC-REG test, 395, 397

ENC-T test, 394, 397, 404, 407, 407n9, 408

encompassing tests, 11, 311, 392, 406, 407, 409–10, 418, 463; under MSE loss, 393–97

Engle, R. F., 219, 246, 280, 284, 290, 303, 305, 306, 308, 367, 459, 478, 479, 493, 494, 513

error correction model (ECM), 478–79

“estimation effect terms,” 51

estimation error, 392, 395–96, 471; OLS (ordinary least squares) estimation error, 321. See also parameter estimation error

estimation windows, 10, 106, 110, 145, 253, 272, 453, 460, 465; expanding estimation window, 376–77, 387; fixed estimation window, 372, 379, 379n2, 387; fixed-proportion estimation window, 379; rolling estimation window, 324, 377–79, 387, 407n9, 409, 424n1, 427, 428

Estrella, A., 267

European Union (EU) commission forecasts, 365

Evans, J. M., 452

expectation-maximization (EM) algorithm, 233

expected loss, 17–18, 41–46, 54, 63, 91, 101

exponential smoothing, 159; and the discounted least squares loss function, 161; equivalence of with ARMA models, 162–64; and the exponentially weighted moving average (EWMA), 159–60, 160n12; extensions of the exponential smoother, 164; unobserved components of, 161–62

exponentially weighted moving average (EWMA), 159, 160, 161, 183, 184

Extended Natural Conjugate (ENC), 198

factor models, 219, 226n4; determining the number of factors to include when forecasting with factor models, 229–32; dynamic factor models (DFMs), 221–23, 235, 238, 239, 333; forecasting with factor models, 220–21. See also factor models, empirical evidence of; factor models, practical issues arising with

factor models, empirical evidence of, 234–36; application of, 236, 238–39, 241; empirical success of factor models, 235; of factor models versus Bayesian VARs, 236

factor models, practical issues arising with, 232; identification and economic interpretation of factors, 232; instability, 232–33; missing observations, 233–34

factors, estimation of, 223; Bayesian methods for, 229; consistency and efficiency in, 225–27; and extraction of factors in the frequency domain, 227–28; maximum likelihood estimation (small N), 223–24; principal components estimation (large N), 224–25

Fair, R. C., 348

Favero, C. A., 320

Federal Reserve Bank of Philadelphia, 322, 496n4, 504

Federal Reserve Bank of St. Louis, 140

Federal Reserve Board, 24, 47, 260; forecasts of, 26. See also Greenbook forecasts

Fernandez, C., 340

finite-order lag polynomials, 223, 223n3

forecast applications, examples of, 31; of a central bank’s decision problem, 31–33; of a directional trading system, 36–37; of a portfolio choice under mean-variance utility, 33–36, 35n14

forecast density, 57, 58, 84, 438

forecast errors, 36, 258n7, 397–98, 397n6; cost of (e = y − f ), 4; covariance between, 28; loss functions that depend only on forecast errors, 20; negative forecast errors, 364n7; and trade-offs, 13, 16; vector of, 28, 28n11

forecast evaluation, 4, 5, 9, 348, 371, 441–42, 486; development of evaluation methods, 10; evaluation of distributional forecasts, 11, 347–48; evaluation of interval forecasts, 440–41; evaluation of multicategory forecasts, 438–40; informal forecast evaluation methods (scatterplots and time-series graphs), 348–50, 352; overview of, 10. See also individual forecasts, evaluation of; multiple forecasts, evaluation of

forecasters, 4, 13–14, 37, 243, 254; the forecaster’s problem as a decision problem, 5, 40; objective of, 14, 39, 41; weather forecasters, 13

forecasting, in economics and finance, 3; the “art” of forecasting, 5; assumption of symmetrical loss in economic forecasting, 17; binary forecasting, 14n1; concern of over forecasting models, 3; in a data-rich environment, 218–20, 243; decision-theoretic approach to, 3–4, 6, 12; forecast combinations, 9; forecasting with panel data, 241–42; goal of, 57; macroeconomic forecasting, 221n1; properties of a good forecast, 10; real-time forecasting methods, 11; regional forecasting, 203; scenario forecasting, 210; standard approaches to forecasting, 6; time-series forecasting, 241. See also factor models; forecasting models; forecasting models, classical estimation of forecasting optimality; forecasting problem, the; model (parameter) instability; nonstandard data, forecasting of; trending variables

forecasting models: construction of, 4, 57, 64, 64n1; forecasting model with a deterministic trend, 469; identifying a forecasting model for a given time series, 140; kernel estimation of, 245–46; nonparametric forecasting models, 40, 244–45, 277, 297–300; “observation driven models,” 446; parametric forecasting models, 40, 277; predictive performance of a single model, 11; semiparametric forecasting models, 40, 277; sets of models compared to a single model, 3; standard forecasting models, 7. See also classical forecasting models; factor models; forecasting models, classical estimation of forecasting optimality; multivariate forecasting models, and trending variables; nested models; nonlinear univariate models; random walk models; sieve models, estimation of; trending variables

forecasting models, classical estimation of forecasting optimality, 63–64, 74–75; estimation based on functions other than the loss function, 69–71; estimation based on penalized loss functions, 71–73; loss-based estimators, 64–68. See also optimality tests; plug-in estimators

forecasting problem, the, 8, 9, 13, 15, 18, 25, 37, 45, 48, 53, 66, 68, 74, 78, 269, 274, 355, 386; basic density forecasting problem, 274; basic form of, 166; binary forecasting problem, 268, 423, 425; and the context of decision theory, 40; as a decision problem, 5; evaluation of forecast methods as central to, 9; general forecasting problem, 39; point forecasting problem, 338; real-time forecasting problem, 373; timing of, 39

forecasts, 15n3; biased forecasts, 377; “broken clock” forecasts, 112; closeness of the forecast to the outcome, 41; cointegration between forecasts and outcomes, 489; conditional forecasts, 210–12; consistent ranking of forecasts with measurement errors in the outcome, 27; desirable properties of, 347–48; feedback effects of, 15n2; Greenbook forecasts, 38; nearest neighbor forecasts, 246; viewed as a signal in a strategic game, 13; volatility forecast comparisons, 27. See also binary forecasts; break process, forecasts that model the; density forecasts; distributional forecasts; forecast applications, examples of

Franses, P. H., 166, 169

FRED database, 140

Friedman, J., 256, 258n7

Fu, W., 119

Fuller, W. A., 474

functional central limit theory (FCLT), 515

Gallant, A. R., 298, 299, 300

Gallo, G., 290

Gargano, A., 72, 330–32, 342

Garrote method, 72

Gaussian density, 299; comparison of Gaussian density forecasts, 430, 431, 432, 438

Gaussian distribution, 83, 282, 430

Gaussian likelihood estimation, 223

Gaussian priors, 229

Gaussian series: computation predictive density for an i.i.d. Gaussian series, 77–78; forecasting the mean of an i.i.d. Gaussian series under Linex loss, 78; forecasting the mean of an i.i.d. Gaussian series under MSE loss, 78, 79

Gaussian shocks, 211

generalized autoregressive conditional heteroscedasticity (GARCH) models, 279, 280–85, 290, 299, 304, 305, 338, 367, 427, 428, 437, 494; the ARMA-GARCH model, 67; the EGARCH model, 284; estimation of GARCH parameters based on loss function, 67, 282–83; probability integral transform (PIT) score for GARCH(1, 1) process, 434–35; refinements to GARCH models, 283–87

generalized least squares (GLS), 242, 472, 473

Genre, V., 342

Geoum, I. S., 13

Geweke, J., 77, 84, 85, 194n4, 219, 267, 338

Ghent, A. C., 209

Ghysels, E., 155n10, 157, 287, 501, 502

Giacomini, R., 382, 392, 395, 399, 400–402, 403, 420, 424n1, 425–26, 451, 465

Giannone, D., 85, 200, 201, 202, 229, 235, 236, 498, 499, 500, 500n5, 501

Gibbs sampler, 84, 198, 200, 212, 267, 458; least squares regression and the Gibbs sampler, 84

Glosten, L. R., 285

Gneiting, T., 426, 430, 438

González, A., 170

González-Rivera, G., 284n4, 438n5

Goyal, A., 106, 108, 140, 414

Granger, C. W. J., 15, 15n4, 16, 17, 25, 46, 47, 74, 141, 185n3, 262n2, 285n5, 314, 317, 319, 320, 321, 326, 330, 357, 366, 397, 420n13, 448; Granger causality tests, 187, 190–91, 463, 464, 465; the Granger representation theorem, 478–79; the three Granger properties, 16, 20, 21

Gray, S. F., 179

“Great Moderation,” the, 445

Greenbook forecasts, 38, 349–51, 352, 361

Greenspan, Alan, 275

Grenander, U., 246

gross domestic product (GDP), 218, 322, 361, 445, 499, 501; advance estimates of, 496; GDP growth in the United States, 179, 200, 323–24, 350, 362, 490

Guidolin, M., 179, 320n4

Gunther, T. A., 423, 435n3

Hadri, K., 479

Haldrup, N., 179

Halling, M., 459

Hamill, T. M., 438

Hamilton, J. D., 143, 166, 172, 173, 174, 181, 182, 183, 191, 505

Hannan, E. J., 96, 98

Hannan-Quinn criterion, 96, 119

Hansen, B.E., 246, 293, 302, 329–30

Hansen, P. R., 27, 280, 288, 289, 290, 362, 404, 405, 408, 411n11, 412, 413, 415, 416, 417, 418, 419

Harris, D., 493

Harrison, J., 459

Harrison, P., 300

Hartmann, P., 200

Harvey, D. S., 394, 396, 398, 400n7, 407

Hastie, T., 72, 256

Hendry, D. F., 28, 94n4, 95, 311n1, 320, 321, 392, 393, 394, 395, 461, 462

Henkel, S. J., 402

heteroscedasticity, 199n7, 278, 366, 503; conditional heteroscedasticity, 95, 408; cross-sectional heteroscedasticity, 230. See also autoregressive conditional heteroscedasticity (ARCH) models; generalized autoregressive conditional heteroscedasticity (GARCH) models

Hjort, N. L., 329

Hodrick–Prescott (HP) filter, 511

Hoel, P. G., 392

Hoeting, J. A., 339, 340, 341

Hoffman, D. L., 25

Holt-Winters procedure, 158–59

homogeneity, 16

Hong, H., 13

Hong, Y., 436n4

Hoover, K. D., 96

Hoque, A., 363

Hubrich, K., 170, 200, 254n6, 311n1, 411n11

Hurvich, C. M., 483

Hyndman, R.

INAR(1) model

INAR(p) model

individual forecasts, evaluation of; evaluation of aggregate versus disaggregate forecasts; formal and informal methods of; and in-sample forecast evaluation; and out-of-sample (OoS) asymptotics for rationality tests; and the production of “small” or “large” expected losses; and pseudo out-of-sample (OoS) forecasting methods; the sampling distribution of average losses. See also out-of-sample (OoS) average loss, conducting inference on; out-of-sample (OoS) forecasts, simulation of

inflation; inflation forecasts. See also inflation rate

inflation rate; definition of; forecasting the quarterly CPI inflation rate; inflation rate series. See also three-variable system (CPI inflation rate, unemployment rate, three-month T-bill rate), empirical example of

information criteria (IC); AIC and BIC for linear regressions; Akaike

information criterion (AIC); Bayes/Bayesian information criterion (BIC); different strategies employed by IC to trade-off fit against parsimony; Hannan-Quinn criterion; Mallows ${\mathrm{C}}_P$ criterion; model selection in linear regressions based on Mallows ${\mathrm{C}}_P,$

information sets

Ingram, B. F.

Inoue, A.

interest rates; modeling the term structure of interest rates

International Monetary Fund (IMF); forecasts of individual countries’ budget deficits

interval/quantile forecasts; estimation of

irrepresentability condition, the

Ito, T.

Jansson, M.

Jenkins, G. M.

Jordà, O.

Kadiyala, K. R.

Kalman filter; basic setting of; and the construction of coincident indicators; derivation of; and estimating parameters; Kalman filter equations; Kalman smoothing; and missing observations

Kalman smoother

Kandel, S.

Kapetanios, G.

Karlsson, S.

Kauppi, H.

Kemp, G. C.

Kendall, M. G.

Kilian, L.

Kim, C.-J.

Kim, T.-H.

Kim, Y. H.

Kinal, T.

kitchen sink model/method

Knight, K.

Koenker, R.

Kolmogorov–Smirnov (KS) test

Komunjer, I.

Koop, G.

Koopman, S. J.

Korajczyk, R. A.

Korobilis, D.

Kourtellos, A.

Krolzig, H.-M.

Kubik, J. D.

Kuipers score

Kullback-Leibler (KL)

Lagrange Multiplier (LM) tests

Lahiri, K.

Lasso (least absolute shrinkage and selection operator) method; and model selection

Laster, D.

Laurent, S.

“least bad” method

least squares: discounted least squares loss function; generalized least squares (GLS); least squares coefficients in a projection of $y_{t+1}.$ ; least squares estimates under absolute error loss; least squares regression and the Gibbs sampler; partial least squares (PLS). See also ordinary least squares (OLS)

Lee, J.-H.

Leeb, H.

Leitch, G.

Leng, C.

Leung, M. T.

leverage effect

Ley, E.

Leybourne, S.

L’Hôpital’s rule

Lieli, R. P.

Likelihood Ratio (LR)

Lin, J.-L.

Lin, Y.

linear regression; AIC and BIC for linear regressions; Bayesian linear regression models; linear regression models; linear regression models with normal priors; linear regression with a single break; local linear regression models; model selection in linear regressions based on Mallows ${\mathrm{C}}_P.$ ; multicollinearity in linear regression; nested linear regression models

Linex loss; Bayes risk under Linex loss; optimal forecast under

lin-lin loss. See piecewise linear loss (lin-lin loss)

Litterman, R. B.; Litterman priors (the “Minnesota priors”)

Liu, P. C.

Livingston Survey

long-horizon returns

loss decomposition methods

loss functions; for all forecasting problems; decision-maker’s loss function; game-theoretical models; Patton’s definition of; specification of ignored in economic forecasting; “standard,”; symmetry of. See also loss functions, construction and specification of; loss functions, specific; volatility, and density forecasting: role of the loss function in

loss functions, construction and specification of; bounded; bowl-shaped; common properties of; constructing a loss function; expected loss; loss functions not based on expected loss; loss function viewed as the negative of a utility function; loss functions that take the form of profit functions; unbounded

loss functions, specific; absolute error loss; Bregman loss; binary loss; EKT loss; level- and forecast-dependent; Linex; loss functions that depend only on forecast errors; loss functions that depend on other state variables; multivariate loss; multivariate quadratic error loss (multivariate MSE loss); piecewise asymmetrical loss. See also mean average percentage error (MAPE) loss

Low, S. A.

Lunde, A.

Lütkepohl, H.

Machina, J.

macroeconomic data

macroeconomic forecasting

macroeconomic time series

Maddala, G. S.

Madigan, D.

Maekawa, K.

Mallows, C. L.

Mallows-Quinn criterion

Magnus, J. R.

Manganelli, G.

Marcellino, M.

Marcucci, J.

Mariano, R. S. See also Diebold-Mariano (DM) test

market timing

Markov switching models (regime switching models), 7, 172–73, 200, 295; and the break process; forecasting with, 173–76; multistep forecasts for the two-state Markov switching process, 175–76; refinements to the model, 177–79; two-state regime switching models, 176–77

Marriott, F.

Martin, G. M.

Martin, J. S.

martingale difference sequences

Massengill, H. E.

maximum likelihood estimation (MLE). See also quasi-maximum likelihood estimator (QMLE)

maximum utility (MU) estimators

McCabe, B. P.

McConnell, M. M.

McCracken, M. W.

McCurdy, T. H.

McFadden, D.

McKenzie, E.

McNeil, A.

mean absolute error (MAE) loss

mean average percentage error (MAPE) loss

mean squared error (MSE) loss, 9, 20, 21–22, 23, 24, 28, 37, 42–43, 45–46, 63, 82, 83, 121, 140, 183, 191, 202, 245, 258, 265, 316, 320, 324, 333, 364, 368, 374, 404, 427, 453–54, 463, 473, 474, 486–88; Bayes risk under MSE loss; best average versus best current model under MSE loss; of a forecast (or fitted value), 9; forecast efficiency tests under MSE loss; forecast encompassing tests under MSE loss; inverse MSE weighting; minimizing MSE results; MSE evaluation of asymmetric quadratic loss; MSE loss given the observed history of the outcome; MSE loss for linear forecasting models; MSE loss for sample estimate of the mean; MSE loss for shrinkage estimators; MSE population values; MSE with misspecified trend; MSE with near-unit root process; optimal forecast under MSE loss; orthogonality regressions under MSE loss; of out-of-sample forecasts, 10; plot of against Linex loss; and the recentered test statistic; squared error loss as a homogenous class; tests of equivalent expected loss under MSE loss; unconditional MSE loss. See also combination weights, estimation of: forecast of under MSE loss; optimal forecast combinations: optimal combinations under MSE loss

Meese, R. A.

M-estimation; based on loss function; under lin-lin loss

Mikkelsen, H. O.

Min, C.-k.

Mincer, J. A.

Mincer–Zarnowitz regression

Minnesota priors

Mishkin, F. S.

mixed data sampling (MIDAS) methods model combination; and complete subset regressions; empirical example of; risk for

model (parameter) instability; and “breakdown” in forecasting performance; breaks and forecasting performance (linear regression model with a single break); and instability of factor models; limitations of in-sample tests for model instability; for models with multiple breaks; for models with a single break; sequential methods for estimating the time of the break and for additional breaks; and shifts in the parameters of the data-generating process; suggested optimal tests for; using a reverse-ordered CUSUM squared (ROC) break test. See also breaks, ad hoc methods for dealing with; break process, forecasts that model the

model selection; “best” model selection; model selection searches; through cross validation; trade-offs in model selection. See also Bagging; information criteria (IC); Lasso; model selection methods

model selection methods: “in-sample” methods; “out-of-sample” (OoS) methods; properties of model selection procedures. See also risk: for model selection methods (Monte Carlo simulations)

Mohanty, S.

monotonicity

Monte Carlo Markov Chain (MCMC)

Monte Carlo simulations/experiments/ methods/estimates; for an AR(1) model; examination of break tests through Monte Carlo simulations; and the Reality Check test statistic. See also Monte Carlo Markov Chain (MCMC); risk: for model selection methods (Monte Carlo simulations)

Morgan, W.

Morgan-Granger-Newbold regression test

Moriera, M. J.

Müller, U. K. See also qLL test (of Elliott and Müller) 11:

multicollinearity

multiple forecasts, comparison of; the Giacomini-White approach; the Giacomini-White approach and the conditional test of forecasting performance; in-sample versus out-of-sample (OoS) comparison; and testing for superior predictive ability (SPA). See also performance, comparison of across nested models

multiple forecasts, evaluation of; forecast encompassing tests; forecast encompassing tests under MSE loss; and tests of equivalent expected loss (test of loss equivalence under MSE loss and the Diebold-Mariano test)

multivariate forecasting models, and trending variables; empirical example of; and multivariate long-run forecasts multivariate volatility models

Muth, J. F.

Nadaraya, E.

Nardari, F.

NASDAQ

Nason, J. A.

Nelson, C. R.

Nelson, D. B.

nested models; comparisons of. See also performance, comparison of across nested models

Newbold, P.

Newey, W. K.

Ng, S.

Nielsen, M. Ø.

Nolan, D.

nonlinear parametric models; parametric versus nonparametric estimation approaches

nonlinear univariate models; empirical comparisons; forecasting with linear versus nonlinear models

nonlinearity: nonlinear dynamics, 7; nonlinear optimization, 223, 251, 252. See also nonlinear parametric models; univariate nonlinear prediction models

nonparametric model, 40, 244–45

nonstandard data, forecasting of; forecasting count data; forecasting durations; parametric models for count data; real-time data

Nordhaus, W. D., 355, 368

nowcasting

Nyblom, J., 449

Nychka, D. W., 298, 299

Olivetti, C., 486

Onatski, A., 231

“one-shot” problem, 19

opinion pool, 337, 337n11

optimal forecast combinations; optimal combinations under Linex loss; optimal combinations under MSE loss

optimality tests: optimality tests that do not rely on measuring the outcome; under known loss; with known loss shape but unknown parameters

ordinary least squares (OLS), 51, 53, 53n7, 152, 229, 242, 322, 333, 448, 454, 455, 456, 463–64, 473, 482–83; OLS estimation error; OLS estimation of VARs; OLS regression; OLS residuals; recursive OLS; restricted OLS

Organization for Economic Cooperation and Development (OECD), 24

Ornstein-Uhlenbeck processes, 469n2

orthogonality, 355, 357, 361, 363, 381, 386, 387, 388, 393, 400, 413, 489

Ottaviani, M., 13

out-of-sample (OoS) asymptotics for rationality tests

out-of-sample (OoS) average loss, conducting inference on; and West’s Monte Carlo results; and West’s results under squared error loss; and West’s theorem

out-of-sample (OoS) forecasts: in-sample versus out-of-sample (OoS) comparison; mean squared error (MSE) loss of, 10; pseudo out-of-sample forecasts; reasons for using. See also out-of-sample (OoS) forecasts, simulation of

out-of-sample (OoS) forecasts, simulation of; and the expanding estimation window; and exponentially declining weights; and the fixed estimation window; and the fixed-proportion estimation window; and the rolling estimation window

overparameterized models, 10

Owyang, M.T., 28, 29

Pagan, A., 246

panel data, forecasting with

parameter estimation error, 7, 35n14, 49, 64, 71, 85, 110, 154, 171, 187, 203, 217, 236, 343, 363, 382–83, 384, 395, 407; effects of; reduction of, 75

parameter estimation models, 7, 8

parametric density models, 40, 292; and mixtures of normal; non-Gaussian density model; parametric versus nonparametric estimation approaches

Patton, A. J., 26, 27, 29, 307, 308, 358n4, 362, 363, 365, 366, 367, 368, 427; definition of a loss function, 27

Pauly, P., 311, 318, 462

Paye, B. S., 445

Perez, S. J., 96

Perez-Quiros, G., 445

performance, comparison of across nested models; complications arising from nested models; empirical application of; and the finite sample behavior of tests; and the recentered test statistic

Perron, P., 449, 455, 455n6, 456, 475

persistent regressors; empirical application of; and results for multivariate long-run forecasts

Pesaran, M. H., 25, 85, 95n5, 112, 166, 192, 194, 235, 262n2, 272, 363, 377, 379n2, 438, 439, 440, 452–53, 454, 457, 465

“peso” problem, the

Pettenuzzo, D., 85, 200, 457

Phillips, P. C., 152, 153, 473, 476, 477, 485

Pick, A., 192, 235

piecewise linear loss (lin-lin loss), 21–22, 23, 58, 67

Ploberger, W., 449

plug-in estimators, 68, 467; mean-variance investor’s use of; plug-in estimators based on the forecaster’s loss function

point forecasts, 3, 8, 14, 29, 39, 40, 58, 78, 80, 83, 236, 260, 261, 271, 273, 301, 308, 323, 336, 337, 422, 423, 425, 426, 465; accurate forecasts; as an application of decision theory, 4; Bayesian; criticism of; generating; inferior. See also point forecasts, optimal

point forecasts, optimal, 41, 62; as conditional on future variables; construction of; dependence of on conditioning variables; and expected loss; interpretation of forecast optimality; optimal forecasts minimizing the average conditional loss

Poisson models, 12, 289, 490, 491, 492, 493, 495

Politis, D. N., 107

Pope, J., 151

portfolio turnover, 35n14

posterior distribution, 55, 58, 60–61, 80, 81–82, 83, 84, 88, 197, 198, 199, 205, 208, 459

Pötscher, B. M., 90

Potter, S., 71, 267

Potter, S. M., 85, 457

Powell, J. L., 23

Powell, O., 340, 340n12

predictive accuracy, comparison of utility-based and statistical measures of

predictive density, 40, 58, 83, 84; computation of for an i.i.d. Gaussian series

probability integral transform (PIT), and density forecasts

projection pursuit regression, 252

Prüfer, P., 340, 340n12

pseudo-true value, 46, 46n5, 67, 381, 382, 411, 467

QLIKE loss, 27

qLL test (of Elliott and Müller), 450

Qu, Z., 455

quadratic loss, 23, 32, 33, 170, 196, 202, 303, 385; MSE evaluation of asymmetric quadratic loss

Reichlin, L., 85, 200, 201, 202, 229, 235, 236, 498, 499, 500, 500n5, 501

RiskMetrics, 160n12

Robertson, J. C., 203

Rogoff, K., 348

quadratic scoring (QS), 338

Ridge regression, 229, 238

Ridge estimators, 72, 81–83

rolling regressions, 396, 402, 460

Quah, D., 219

risk, 40, 373–74; Bayes risk; comparing classical and Bayes risk under Linex loss with Gaussian data; of a forecasting method; and the information set; minimax and average risk; minimizing of; for model selection methods (Monte Carlo simulations); risk functions; risk for a multivariate regression model; risk for subset regressions

resolution, 354–55, 422, 428, 429, 431–32, 441

Romano, J. P., 415

Ridge forecasts, 238

quantile forecasts. See interval/quantile forecasts

quasi-maximum likelihood estimator (QMLE), 70, 283, 291, 291n10, 299, 458n9; estimation of with normal distribution; QMLE for tick-exponential family

Rombouts, J.V., 29, 305

Quinn, B. G., 96, 98

Racine, J.

root mean squared error (RMSE) values

Ramanathan, R.

Raftery, A. E.

recession

Rossi, A. G.

random walk models; moving average (MA) representation of the random walk model; random walk breaks; time-varying parameter (TVP) random walk model

Receiver Operator Characteristic (ROC) curve

recursive updates

Rossi, B.; Rossi tests

Rudebusch, G. D.

regime switching models. See Markov switching models (regime switching models)

Ravazzolo, F.

“real-time” performance

Reality Check (White’s bootstrap Reality Check approach); and model building

Ranjan, R.

Russell, J. R.

Ratner, J.

Rapach, D. E.

rationality; forecast rationality tests

Saikkonen, P.

Sala, L.

Sampson, M.

Sarantis, N.

Sargent, T. J.

SARIMA models

Satchell, S.

Scharfstein, D. S.

Schaumburg, E.

Schervish, M. J.

Schorfheide, F.

Schwarz, G.

Scotti, C.

seemingly unrelated regressions (SUR)

Sekhposyan, T.

Senyuz, Z.

sequential hypothesis testing; illustration of the general-to-specific modeling approach; illustration of the specific-to-general modeling approach; the “LSE” general-to-specific approach to; methods of (specific-to-general and general-to-specific); variations of sequential tests

sequential out-of-sample prediction, evaluation of

Shao, J.

Shephard, N.

Sheppard, K.

Shibata, R.

Shiller, R. J.

shocks: effect of a shock on future average volatility; Gaussian shocks; idiosyncratic shocks; positive shocks

shrinkage estimators/estimation; generalized shrinkage methods

Shuford Jr, E. H.

sieve models, estimation of; and g functions; and polynomials; and projection pursuit regression; and splines. See also artificial neural networks (ANN)

sign test

Sims, C.

Sims, C. A.

Sinko, A.

Skouras, S.

Small, D.

Smets, F.

Smith, A. D.

Smith, J.

Song, F.

Sørensen, P. N.

sparse models

spherical scoring rules

splines

Springborn, M.

squared error loss. See mean squared error (MSE) loss

Srinivasan, V.

Stambaugh, R. F.

Stark, T.

state probabilities, recursive updates of

statistics, “prequential” approach to

Steel, M. F.

Stein, J. C.

Stekler, H. O.

Stepwise testing methods

stochastic breaks

stochastic conditional duration model

stochastic discount factor

stochastic drift

stochastic process

stochastic variations

stochastic volatility models

Stock, J. H.

stock prices, log-linearized present value model for

stock returns; daily stock returns; excess stock returns; prediction of stock returns through dividend yield. See also stock returns, forecasting of

stock returns, forecasting of; application of Bayesian model averaging (BMA) to; benchmark used in; and block size; and the Diebold-Mariano test; and the economic measures of forecast performance; and the out-of-sample R2 measure; point forecasting of stock returns; and recursive inclusion frequencies; the statistical measures of forecast performance; use of the Lasso method in; variables used in

Stone, C. J.

Straumann, D.

Stuart, A.

Sullivan, R.

superior models: and the choice of sample split; identification of; testing for superior predictive ability/accuracy (SPA)

Survey of Professional Forecasters

Svensson, L. E.

Swanson, N. R.

symmetry; assumption of symmetrical loss in economic forecasting; assumption of symmetry for the cost function

Tallman, E. W.

Tanner, J. E.

Tauchen, G.

Tay, A. S.

Taylor, M. P.

Taylor expansion

T-bills. See also three-variable system (CPI inflation rate, unemployment rate, three-month T-bill rate), empirical example of three-month T-bill rate),empirical example

Teräsvirta, T.

Tetlow, R. J.

Theil, H.

three-variable system (CPI inflation rate, unemployment rate, three-month T-bill rate), empirical example of

Tibshirani, R.

Timmermann, A.

Treasury bill futures contracts

trending variables; definition of a trending variable; examples of (forecasting model with a deterministic trend; MSE with misspecified trend); expected loss with trending variables. See also multivariate forecasting models, and trending variables; univariate forecasting models, and trending variables

Trueman, B.

Tsay, R. S.

Turner, J. L.

Tzavalis, E.