Single equation methods (econometrics)

A variety of methods are used in econometrics to estimate models consisting of a single equation. The oldest and still the most commonly used is the ordinary least squares method used to estimate linear regressions.
A variety of methods are available to estimate non-linear models. A particularly important class of non-linear models are those used to estimate relationships where the dependent variable is discrete, truncated or censored. These include logit, probit and Tobit models.
Single equation methods may be to applied time-series, cross section or panel data.

Economic model

In economics, a model is a theoretical construct that represents economic processes by a set of variables and a set of logical and/or quantitative relationships between them. The economic model is a simplified framework designed to illustrate complex processes, often but not always using mathematical techniques. Frequently, economic models use structural parameters. Structural parameters are underlying parameters in a model or class of models.A model may have various parameters and those parameters may change to create various properties.

models

When we do any kind of classification/categorization, we need an "according to" or "by" clause. So according to whether all the model variables are deterministic, economic models can be classified as stochastic or non-stochastic models; according to whether all the variables are quantitative, economic models are classified as discrete or continuous choice model; according to the model's intended purpose/function, it can be classified as quantitative or qualitative; according to the model's ambit, it can be classified as a general equilibrium model, a partial equilibrium model, or even non-equilibrium model; according to the economic agent's characteristics, models can be classified as rational agent models, representative agent models etc.

• Stochastic models are formulated using stochastic processes. They model economically observable values over time. Most of econometrics is based on statistics to formulate and test hypotheses about these processes or estimate parameters for them. 

• Non-stochastic mathematical models may be purely qualitative (for example, models involved in some aspect of social choice theory) or quantitative (involving rationalization of financial variables, for example with hyperbolic coordinates, and/or specific forms of functional relationships between variables). In some cases economic predictions of a model merely assert the direction of movement of economic variables, and so the functional relationships are used only in a qualitative sense: for example, if the price of an item increases, then the demand for that item will decrease. For such models, economists often use two-dimensional graphs instead of functions.

• Qualitative models - Although almost all economic models involve some form of mathematical or quantitative analysis, qualitative models are occasionally used. One example is qualitative scenario planning in which possible future events are played out. Another example is non-numerical decision tree analysis. Qualitative models often suffer from lack of precision.

At a more practical level, quantitative modelling is applied to many areas of economics and several methodologies have evolved more or less independently of each other. As a result, no overall model taxonomy is naturally available. We can nonetheless provide a few examples which illustrate some particularly relevant points of model construction.

• An accounting model is one based on the premise that for every credit there is a debit. More symbolically, an accounting model expresses some principle of conservation in the form

algebraic sum of inflows = sinks − sources

This principle is certainly true for money and it is the basis for national income accounting. Accounting models are true by convention, that is any experimental failure to confirm them, would be attributed to fraud, arithmetic error or an extraneous injection (or destruction) of cash which we would interpret as showing the experiment was conducted improperly.

• Optimality and constrained optimization models - Other examples of quantitative models are based on principles such as profit or utility maximization. An example of such a model is given by the comparative statics of taxation on the profit-maximizing firm. The profit of a firm is given by

where p(x) is the price that a product commands in the market if it is supplied at the rate x, xp(x) is the revenue obtained from selling the product, C(x) is the cost of bringing the product to market at the rate x, and t is the tax that the firm must pay per unit of the product sold.

The profit maximization assumption states that a firm will produce at the output rate x if that rate maximizes the firm's profit. Using differential calculus we can obtain conditions on x under which this holds. The first order maximization condition for x is

Regarding x is an implicitly defined function of t by this equation (see implicit function theorem), one concludes that the derivative of x with respect to t has the same sign as

which is negative if the second order conditions for a local maximum are satisfied.

Thus the profit maximization model predicts something about the effect of taxation on output, namely that output decreases with increased taxation. If the predictions of the model fail, we conclude that the profit maximization hypothesis was false; this should lead to alternate theories of the firm, for example based on bounded rationality.

Borrowing a notion apparently first used in economics by Paul Samuelson, this model of taxation and the predicted dependency of output on the tax rate, illustrates an operationally meaningful theorem; that is one which requires some economically meaningful assumption which is falsifiable under certain conditions.

• computationally much harder to deal with and harder to use as tools for qualitative analysis. For this reason, macroeconomic models usually lump together different variables into a single quantity such as output or price. Moreover, quantitative relationships between these aggregate variables are often parts of important macroeconomic theories. This process of aggregation and functional dependency between various aggregates usually is interpreted statistically and validated by econometrics. For instance, one ingredient of the Keynesian model is a functional relationship between consumption and national income: C = C(Y). This relationship plays an important role in Keynesian analysis.elow

Quantitative vs. Qualitative models

A quantitative model is designed to produce accurate predictions, without elucidating the underlying dynamics. On the other hand, a qualitative model aims to explain these dynamics without necessarily fitting empirical data or informing accurate predictions. Interest rate parity can be deemed a qualitative model in this sense: though it generally fails to fit exchange rate data as well as higher-powered statistical forecasting models, it offers an intuitive interpretation of the exchange rate and its relation to foreign and domestic interest and inflation rates. Views on the relative merits of qualitative and quantitative models vary across the profession: Milton Friedman can be viewed as having advocated a qualitative approach, while Ronald Coase worried that "if you torture the data long enough, it will confess;" Prospect theory as proposed by Daniel Kahneman(a Nobel prize winner) is more quantiative, while rational agent models are more qualitative.