|
|
| Line 1: |
Line 1: |
| The '''parameter identification problem''' is a problem which can occur in the estimation of multiple-equation [[econometric]] models where the equations have variables in common. | | The author is known by the name of Numbers Lint. Puerto Rico is where he's been living for years and he will by no means transfer. Since she was 18 she's been operating as a receptionist but her promotion by no means arrives. Playing baseball is the hobby he will never quit performing.<br><br>Here is my web blog ... [http://www.siccus.net/blog/15356 http://www.siccus.net/] |
| | |
| More generally, the term can be used to refer to any situation where a statistical model will invariably have more than one set of parameters which generate the same distribution of observations.
| |
| | |
| ==The standard example, with two equations==
| |
| Consider a linear model for the [[supply and demand]] of some specific good. The quantity demanded varies negatively with the price: a higher price decreases the quantity demanded. The quantity supplied varies directly with the price: a higher price increases the quantity supplied.
| |
| | |
| Assume that, say for several years, we have data on both the price and the traded quantity of this good. Unfortunately this is not enough to identify the two equations (demand and supply) using [[regression analysis]] on observations of ''Q'' and ''P'': of course one can not estimate a downward slope ''and'' an upward slope with one linear regression line involving only two variables. Additional variables can make it possible to identify the individual relations.
| |
| | |
| [[File:Supply and demand.png|325px|right|Supply and demand]]
| |
| | |
| In the graph shown here, the supply curve (red line, upward sloping) shows the quantity supplied depending positively on the price, while the demand curve (black lines, downward sloping) shows quantity depending negatively on the price and also on some additional variable ''Z'', which affects the location of the demand curve in quantity-price space. This ''Z'' might be consumers' income, with a rise in income shifting the demand curve outwards. This is symbolically indicated with the values 1, 2 and 3 for ''Z''.
| |
| | |
| With the quantities supplied and demanded being equal, the observations on quantity and price are the three white points in the graph: they reveal the supply curve. Hence the effect of ''Z'' on ''demand'' makes it possible to identify the (positive) slope of the ''supply'' equation. The (negative) slope parameter of the demand equation cannot be identified in this case. In other words, the parameters of an equation can be identified if it is known that some variable does ''not'' enter into the equation, while it does enter the other equation.
| |
| | |
| A situation in which both the supply and the demand equation are identified arises if there is not only a variable ''Z'' entering the demand equation but not the supply equation, but also a variable ''X'' entering the supply equation but not the demand equation:
| |
| | |
| : '''supply:''' <math> Q = a_S + b_S P + cX \, </math>
| |
| | |
| : '''demand:''' <math> Q = a_D + b_D P + d Z \, </math>
| |
| | |
| with positive ''b<sub>S</sub>'' and negative ''b<sub>D</sub>''. Here both equations are identified if ''c'' and ''d'' are nonzero.
| |
| | |
| Note that this is the structural form of the model, showing the relations between the ''Q'' and ''P''. The [[reduced form]] however can be identified easily.
| |
| | |
| ==Estimation methods and disturbances==
| |
| "It is important to note that the problem is not one of the appropriateness of a particular estimation technique. In the situation described [without the ''Z'' variable], there clearly exists ''no'' way using ''any'' technique whatsoever in which the true demand (or supply) curve can be estimated. Nor, indeed, is the problem here one of statistical inference - of separating out the effects of random disturbance. There is no disturbance in this model [...] It is the logic of the supply-demand equilibrium itself which leads to the difficulty." (Fisher 1966, p. 5)
| |
| | |
| ==More equations==
| |
| More generally, consider a linear system of ''M'' equations, with ''M'' > 1.
| |
| <!-- The symbol M is used here to be compatible with the Fisher citation below -->
| |
| | |
| An equation can not be identified from the data if less than <math>M-1</math> variables are excluded from that equation. This is a particular form of the ''order condition'' for identification. (The general form of the order condition deals also with restrictions other than exclusions.) The order condition is necessary but not sufficient for identification.
| |
| | |
| The ''rank condition'' is a necessary and sufficient condition for identification. In the case of only exclusion restrictions, it must "be possible to form at least one nonvanishing determinant of order <math>M-1</math> from the columns of ''A'' corresponding to the variables excluded a priori from that equation" (Fisher 1966, p. 40), where ''A'' is the matrix of coefficients of the equations. This is the generalization in matrix algebra of the requirement "while it does enter the other equation" mentioned above (in the line above the formulas).
| |
| | |
| ==Related use of the term==
| |
| | |
| In engineering language, the term "parameter identification" is used to indicate a more general subject, which is roughly the same as [[estimation]] in [[statistics]].
| |
| | |
| ==See also==
| |
| * [[Instrumental variable#Identification]]
| |
| * [[Errors-in-variables model#Linear model]], discussing identifiability
| |
| * [[Observational equivalence]]
| |
| * [[Identifiability]]
| |
| * [[System of linear equations]]
| |
| * [[Simultaneous equations]]
| |
| | |
| {{More footnotes|date=December 2009}}
| |
| | |
| ==References==
| |
| * {{cite journal |last=Koopmans |first=Tjalling C. |title=Identification problems in economic model construction |journal=Econometrica |volume=17 |issue=2 |date=1949 |pages=125–144 | doi=10.2307/1905689 |publisher=The Econometric Society |jstor=1905689 }} ("A classic and masterful exposition of the subject", Fisher 1966, p. 31)
| |
| | |
| * {{cite book | first=Franklin M. | last=Fisher | year=1966 | title=The Identification Problem in Econometrics | isbn=0-88275-344-4}}
| |
| * See also any modern book on [[econometrics]].
| |
| | |
| [[Category:Econometrics]]
| |
The author is known by the name of Numbers Lint. Puerto Rico is where he's been living for years and he will by no means transfer. Since she was 18 she's been operating as a receptionist but her promotion by no means arrives. Playing baseball is the hobby he will never quit performing.
Here is my web blog ... http://www.siccus.net/