Estimating regression models of finite but unknown order

This paper considers some problems associated with estimation and inference in the normal linear regression model

$\text{y}{}_{\mathrm{t}}\text{=}\text{∑}\text{j=1}\text{m0}\text{β}{}_{\mathrm{j}}\text{x}{}_{\mathrm{tj}}\text{+ε}{}_{\mathrm{t}}\text{,}\text{var}\text{(ε}{}_{\mathrm{t}}\text{)=σ}{}^2$

, when $\text{m}{}_0$ is unknown. The regressors are taken to be stochastic and assumed to satisfy V. Grenander's (1954) conditions almost surely. It is further supposed that estimation and inference are undertaken in the usual way, conditional on a value of $\text{m}{}_0$ chosen to minimize the estimation criterion function

$\text{EC(m, T)=}\text{σ}\text{?}{}^2{}_{\mathrm{m}}\text{+ mg(T)}$

, with respect to m, where σ̂²_m is the maximum likelihood estimate of $\text{σ}{}^2$. It is shown that, subject to weak side conditions, if $\text{g(T)}\text{→}\text{a.s.}\text{0}$ and $\text{Tg(T)}\text{→}\text{a.s.}\text{∞}$ then this estimate is weakly consistent. It follows that estimates conditional on the chosen value of $\text{m}{}_0$ are asymptotically efficient, and inference undertaken in the usual way is justified in large samples. When $\text{g(T)}$ converges to a positive constant with probability one, then in large samples $\text{m}{}_0$ will never be chosen too small, but the probability of choosing $\text{m}{}_0$ too large remains positive.The results of the paper are stronger than similar ones [R. Shibata (1976), R.J. Bhansali and D.Y. Downham (1977)] in that a known upper bound on $\text{m}{}_0$ is not assumed. The strengthening is made possible by the assumptions of strictly exogenous regressors and normally distributed disturbances. The main results are used to show that if the model selection criteria of H. Akaike (1974), T. Amemiya (1980), C.L. Mallows (1973) or E. Parzen (1979) are used to choose $\text{m}{}_0$ in (1), then in the limit the probability of choosing $\text{m}{}_0$ too large is at least 0.2883. The approach taken by G. Schwarz (1978) leads to a consistent estimator of $\text{m}{}_0$, however. These results are illustrated in a small sampling experiment.     

An efficient implementation of the Wagner-Whitin algorithm for dynamic lot-sizing

We consider an N-period planning horizon with known demands D_t ordering cost A_t, procurement cost, C_t and holding cost H_t in period t. The dynamic lot-sizing problem is one of scheduling procurement Q_t in each period in order to meet demand and minimize cost.The Wagner-Whitin algorithm for dynamic lot sizing has often been misunderstood as requiring inordinate computational time and storage requirements. We present an efficient computer implementation of the algorithm which requires low core storage, thus enabling it to be potentially useful on microcomputers.The recursive computations can be stated as follows:

$\text{M}{}_{\mathrm{jk}}\text{=A}{}_{\mathrm{j}}\text{+C}{}_{\mathrm{j}}\text{Q}{}_{\mathrm{j}}\text{+}\text{∑}\text{k?1}\text{t=j}\text{H}{}_{\mathrm{t}}\text{∑}\text{k}\text{r=t+1}\text{D}{}_{\mathrm{r}}$

$\text{F}{}_{\mathrm{k}}\text{=}\text{min}\text{1?j?k}\text{[F}{}_{\mathrm{j}}\text{+M}{}_{\mathrm{jk}}\text{];}\text{F}{}_0\text{=0}$

where M_jk is the cost incurred by procuring in period j for all periods j through k, and F_k is the minimal cost for periods 1 through k. Our implementation relies on the following observations regarding these computations:

$\text{M}{}_{\mathrm{j,k}}\text{=A}{}_{\mathrm{j}}\text{+C}{}_{\mathrm{j}}\text{D}{}_{\mathrm{j}}$

$\text{M}{}_{\mathrm{j,k+1}}\text{=M}{}_{\mathrm{jk}}\text{+D}{}_{\mathrm{k+1}}\text{(C}{}_{\mathrm{j}}\text{+}\text{∑}\text{k}\text{t=j}\text{H}{}_{\mathrm{t}}\text{,}\text{k?j}$

Using this recursive relationship, the number of computations can be greatly reduced. Specifically, $\text{3}\text{2}$N² ? $\text{1}\text{2}$N² additions and $\text{1}\text{2}$N² + $\text{1}\text{2}$N multiplications are required. This is insensitive to the data.A FORTRAN implementation on an Amdahl 470 yielded computation times (in 10^?3 seconds) of T = ?.249 + .0239N + .00446N². Problems with N = 500 were solved in under two seconds.     

Optimum stock levels for excess inventory items

Insufficient attention has been focused on the ubiquitous problem of excess inventory levels. This paper develops two models of different complexity for determining if stock levels are economically unjustifiable and, if so, for determining inventory retention levels. The models indicate how much stock should be retained for regular use and how much should be disposed of at a salvage price for a given item.The first model illustrates the basic logic of this approach. The net benefit realized by disposing of some quantity of excess inventory is depicted by the following equation: HOLDING NET = SALVAGE + COST - REPURCHASE - REORDER BENEFIT REVENUE SAVINGS COSTS COSTSThis relationship can be depicted mathematically as a parabolic function of the time supply retained. Using conventional optimization, the following optimum solution is obtained:

$\text{to=}\text{P?P}{}_{\mathrm{s}}\text{+}\text{C}\text{Q}\text{PF}\text{+}\text{Q}\text{2R}$

where t₀ is the optimum time supply retained; P is the per-unit purchase price; P_s is the per-unit salvage price; C is the ordering cost; Q is the usual item lot size; F is the holding cost fraction; and R is the annual demand for the item. Any items in excess of the optimum time supply should be sold at the salvage price.The second model adjusts holding cost savings, repurchase costs, and reorder costs to account for present value considerations and for inflation. The following optimum relationship is derived:

$\text{PFR}\text{2k}\text{?}\text{PFtR}\text{2}\text{e}{}^{\mathrm{?kt}}\text{+}\text{PFQ}\text{2}\text{+}\text{PQ(i?k)+C(i?k)}\text{e}{}^{\mathrm{(i?k)Q/R}}\text{?1}\text{e}{}^{\mathrm{(i?k)t}}\text{-P}{}_{\mathrm{s}}\text{R?}\text{PFR}\text{2k}\text{=0}$

where i is the expected inflation rate and k is the required rate of return. Unfortunately this relationship cannot be solved analytically for t₀; Newton's Method can be used to find a numerical solution.The solutions obtained by the two models are compared. Not surprisingly, the present value correction tends to reduce the economic time supply to be retained, since repurchase costs and reorder costs are incurred in the future while salvage revenue and holding cost savings are realized immediately. Additionally, both models are used to derive a relationship to describe the minimum economic salvage value, which is the minimum salvage price for which excess inventory should be sold.The simple model, which does not correct for time values, can be used by any organization with the sophistication level to use an EOQ. The present value model which includes an inflation correction is more complex, but can readily be used on a microcomputer. These models are appropriate for independent demand items. It is believed that these models can reduce inventory investment and improve bottom line performance.     

A result on the sign of restricted least-squares estimates

When a variable is dropped from a least-squares regression equation, there can be no change in sign of any coefficient that is more significant than the coefficient of the omitted variable. More generally, a constrained least squares estimate of a parameter β_j must lie in the interval $\text{(}\text{β}\text{?}{}_{\mathrm{j}}\text{?V}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}{}_{\mathrm{jj}}\text{|t|,}\text{β}\text{?}{}_{\mathrm{j}}\text{+ V}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}{}_{\mathrm{jj}}\text{|t|)}$ where $\text{β}\text{?}{}_{\mathrm{j}}$ is the unconstrained estimate of $\text{β}{}_{\mathrm{j}}\text{, V}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}{}_{\mathrm{jj}}$ is the standard error of $\text{β}\text{?}{}_{\mathrm{j}}$ and t is the t-value for testing the (univariate) restriction.     

The structure of exactly strongly consistent social choice functions

For ? an anonymous exactly strongly consistent social choice function (ESC SCF) and x an alternative, define $\text{b}{}_{\mathrm{x}}\text{=b(}{}^{\mathrm{f}}\text{)}{}_{\mathrm{x}}$ to be the size of a minimal blocking coalition for x. By studying the correspondence between ? and $\text{{b(}{}^{\mathrm{f}}\text{)}{}_{\mathrm{x}}\text{}}$, we establish the existence, uniqueness and monotonicity of ESC SCF's. We also prove the following conjecture of B. Peleg: A non–constant anonymous ESC SCF depends on the knowledge of every player's full preference profile.     

Uncorrelated residuals from linear models

In the usual linear model $\text{y}\text{=}\text{Xβ}\text{+}\text{u}$, the error vector u is not observable and the vector r of least squares residuals has a singular covariance matrix that depends on the design matrix X. We approximate u by a vector$\text{r}{}^1\text{=}\text{G}\text{(}\text{JA'y}\text{+}\text{Kz}\text{)}$ of uncorrelated ‘residuals’, where G and $\text{(}\text{J, K}\text{)}$ are orthogonal matrices, $\text{A'X}\text{=}\text{0}$ and $\text{A'A}\text{=}\text{I}$, while z is either 0 or a random vector uncorrelated with u satisfying $\text{E(}\text{z}\text{) = E(}\text{J'u}\text{) =}\text{0}$, $\text{V(}\text{z}\text{) = V(}\text{J'u}\text{) = σ}{}^2\text{I}$. We prove that $\text{r}{}^1\text{-}\text{r}$ is uncorrelated with $\text{r}\text{-}\text{u}$, for any such r¹, extending the results of Neudecker (1969). Building on results of Hildreth (1971) and Tiao and Guttman (1967), we show that the BAUS residual vector $\text{r}{}_{\mathrm{h}}\text{=}\text{r}\text{+}\text{P}{}_{\mathrm{<mtext>1</mtext>}}\text{z}$, where P₁ is an orthonormal basis for X, minimizes each characteristic root of $\text{V(}\text{r}{}^1\text{-}\text{u}\text{)}$, while the vector r_b of Theil's BLUS residuals minimizes each characteristic root of $\text{V(}\text{Jr}{}_{\mathrm{a}}\text{-}\text{r}\text{)}$, cf. Grossman and Styan (1972). We find that $\text{tr V(}\text{r}{}_{\mathrm{h}}\text{-}\text{u}\text{) < tr V(}\text{Jr}{}_{\mathrm{b}}\text{-}\text{u}\text{)}$ if and only if the average of the singular values of $\text{P}\text{′}{}_1\text{K}$ is less than $\text{1}\text{2}$, and give examples to show that BAUS is often better than BLUS in this sense.     

Exact and superlative index numbers

The paper rationalizes certain functional forms for index numbers with functional forms for the underlying aggregator function. An aggregator functional form is said to be ‘flexible’ if it can provide a second order approximation to an arbitrary twice diffentiable linearly homogeneous function. An index number functional form is said to be ‘superlative’ if it is exact (i.e., consistent with) for a ‘flexible’ aggregator functional form. The paper shows that a certain family of index number formulae is exact for the ‘flexible’ quadratic mean of order r aggregator function, $\text{(Σ}{}_{\mathrm{i}}\text{Σ}{}_{\mathrm{j}}\text{a}{}_{\mathrm{ij}}\text{x}{}_{\mathrm{i}}\text{r}\text{2}\text{x}{}_{\mathrm{j}}\text{r}\text{2}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>r</mtext>}}\text{,}$ defined by Den and others. For r equals 2, the resulting quantity index is Irving Fisher's ideal index. The paper also utilizes the Malmquist quantity index in order to rationalize the Törnqvist-Theil quantity indexin the nonhomothetic case. Finally, the paper attempts to justify the Jorgenson-Griliches productivity measurement technique for the case of discrete (as opposed to continuous) data.     

The cointegrated vector autoregressive model with general deterministic terms

In the cointegrated vector autoregression (CVAR) literature, deterministic terms have until now been analyzed on a case-by-case, or as-needed basis. We give a comprehensive unified treatment of deterministic terms in the additive model $X_t=\gamma Z_t+Y_t$, where $Z_t$ belongs to a large class of deterministic regressors and $Y_t$ is a zero-mean CVAR. We suggest an extended model that can be estimated by reduced rank regression, and give a condition for when the additive and extended models are asymptotically equivalent, as well as an algorithm for deriving the additive model parameters from the extended model parameters. We derive asymptotic properties of the maximum likelihood estimators and discuss tests for rank and tests on the deterministic terms. In particular, we give conditions under which the estimators are asymptotically (mixed) Gaussian, such that associated tests are ${\chi }^2$-distributed.     

Inequalities for the Gini coefficient of composite populations

Unlike other popular measures of income inequality, the Gini coefficient is not decomposable, i.e., the coefficient G($\text{X}$) of a composite population $\text{X}\text{=}\text{X}{}_1\text{∪…∪}\text{X}{}_{\mathrm{r}}$ cannot be computed in terms of the sizes, mean incomes and Gini coefficients of the components $\text{X}$_i. In this paper upper and lower bounds (best possible for r=2) for G($\text{X}$) in terms of these data are given. For example, $\text{G(}\text{X}{}_1\text{∪…∪}\text{X}{}_{\mathrm{r}}\text{)≧Σα}{}_{\mathrm{i}}\text{G(}\text{X}{}_{\mathrm{i}}\text{)}$, where α_i is the proportion of the pop ulation in $\text{X}$_i. One of the tools used, which may be of interest for other applications, is a lower bound for ∫^∞₀f(x)g(x)dx (converse to Cauchy's inequality) for monotone decreasing functions f and g.     

Approximations of the eigenvalues of the covariance matrix of a first-order autoregressive process

This paper provides explicit estimates of the eigenvalues of the covariance matrix of an autoregressive process of order one. Also explicit error bounds are established in closed form. Typically, such an error bound is given by $\text{ε}{}_{\mathrm{k}}\text{= (}\text{4}\text{(n+1)}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{ρ}{}^2\text{sin}\text{(}\text{kπ}\text{(n+1)}\text{)}$, so that the approximations improve as the size of the matrix increases. In other words, the accuracy of the approximations increases as direct computations become more costly.