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<title>Parameter identification problem</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Parameter identification problem</span></span>

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<p>In <a href="Economics" title="Economics">economics</a> and <a href="Econometrics" title="Econometrics">econometrics</a>, the <b>parameter identification problem</b> arises when the value of one or more <a href="Parameter" title="Parameter">parameters</a> in an <a href="Economic_model" title="Economic model">economic model</a> cannot be determined from observable variables. It is closely related to <a href="Identifiability" title="Identifiability">non-identifiability</a> in <a href="Statistics" title="Statistics">statistics</a> and econometrics, which occurs when a <a href="Statistical_model" title="Statistical model">statistical model</a> has more than one set of parameters that generate the same distribution of observations, meaning that multiple parameterizations are <a href="Observational_equivalence" title="Observational equivalence">observationally equivalent</a>.

</p><p>For example, this problem can occur in the estimation of multiple-equation econometric models where the equations have variables in common.

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<div class="mw-heading mw-heading2"><h2 id="In_simultaneous_equations_models">In simultaneous equations models</h2></div>

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</style><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="Simultaneous_equations_model" title="Simultaneous equations model">Simultaneous equations model</a> and <a href="System_of_linear_equations" title="System of linear equations">System of linear equations</a></div>

<div class="mw-heading mw-heading3"><h3 id="Standard_example,_with_two_equations">Standard example, with two equations</h3></div>

<p>Consider a linear model for the <a href="Supply_and_demand" title="Supply and demand">supply and demand</a> 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.

</p><p>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 <a href="Regression_analysis" title="Regression analysis">regression analysis</a> on observations of <i>Q</i> and <i>P</i>: one cannot estimate a downward slope <i>and</i> an upward slope with one linear regression line involving only two variables. Additional variables can make it possible to identify the individual relations.

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<p>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 <i>Z</i>, which affects the location of the demand curve in quantity-price space. This <i>Z</i> 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 <i>Z</i>.

</p><p>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 <i>Z</i> on <i>demand</i> makes it possible to identify the (positive) slope of the <i>supply</i> 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 <i>not</i> enter into the equation, while it does enter the other equation.

</p><p>A situation in which both the supply and the demand equation are identified arises if there is not only a variable <i>Z</i> entering the demand equation but not the supply equation, but also a variable <i>X</i> entering the supply equation but not the demand equation:

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<dl><dd><b>supply:</b>    <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q=a_{S}+b_{S}P+cX\,}">

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<annotation encoding="application/x-tex">{\displaystyle Q=a_{S}+b_{S}P+cX\,}</annotation>

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</math></span><img src="./4351857231e05b496152c7763c01cf5195a5a184.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.549ex; height:2.509ex;" alt="{\displaystyle Q=a_{S}+b_{S}P+cX\,}" loading="lazy"></span></dd></dl>

<dl><dd><b>demand:</b>   <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q=a_{D}+b_{D}P+dZ\,}">

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<annotation encoding="application/x-tex">{\displaystyle Q=a_{D}+b_{D}P+dZ\,}</annotation>

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</math></span><img src="./4d766561fdb2a28b8c0c818193e22f1d10723d21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.06ex; height:2.509ex;" alt="{\displaystyle Q=a_{D}+b_{D}P+dZ\,}" loading="lazy"></span></dd></dl>

<p>with positive <i>b<sub>S</sub></i> and negative <i>b<sub>D</sub></i>. Here both equations are identified if <i>c</i> and <i>d</i> are nonzero.

</p><p>Note that this is the <a href="Structural_form" class="mw-redirect" title="Structural form">structural form</a> of the model, showing the relations between the <i>Q</i> and <i>P</i>. The <a href="Reduced_form" title="Reduced form">reduced form</a> however can be identified easily.

</p><p>Fisher points out that this problem is fundamental to the model, and not a matter of statistical estimation:

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</style><blockquote class="templatequote"><p>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 <i>Z</i> variable], there clearly exists <i>no</i> way using <i>any</i> 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)</p></blockquote>

<div class="mw-heading mw-heading3"><h3 id="More_equations">More equations</h3></div>

<p>More generally, consider a linear system of <i>M</i> equations, with <i>M</i> > 1.

</p><p>An equation cannot be identified from the data if less than <i>M</i> − 1 variables are excluded from that equation. This is a particular form of the <a href="Order_condition" class="mw-redirect" title="Order condition">order condition</a> 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.

</p><p>The <a href="Rank_condition" class="mw-redirect" title="Rank condition">rank condition</a> is a <a href="Necessity_and_sufficiency" title="Necessity and sufficiency">necessary and sufficient</a> condition for identification. In the case of only exclusion restrictions, it must "be possible to form at least one nonvanishing determinant of order <i>M</i> − 1 from the columns of <i>A</i> corresponding to the variables excluded a priori from that equation" (Fisher 1966, p. 40), where <i>A</i> 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).

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<ul><li><a href="Identifiability" title="Identifiability">Identifiability</a>, the related problem in statistics</li>

<li><a href="Errors-in-variables_model#Linear_model" title="Errors-in-variables model">Errors-in-variables model#Linear model</a></li>

<li><a href="Instrumental_variable" class="mw-redirect" title="Instrumental variable">Instrumental variable#Identification</a></li>

<li><a href="Set_identification" title="Set identification">Set identification</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFFisher1966" class="citation book cs1"><a href="Franklin_M._Fisher" title="Franklin M. Fisher">Fisher, Franklin M.</a> (1966). <i>The Identification Problem in Econometrics</i>. McGraw-Hill. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-88275-344-4</bdi>.</cite></li>

<li><cite id="CITEREFGreenbergWebster1983" class="citation book cs1">Greenberg, Edward; Webster, Charles E. Jr. (1983). "The Identification Problem". <i>Advanced Econometrics : A Bridge to the Literature</i>. New York: John Wiley & Sons. pp. <span class="nowrap">221–</span>241. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-471-09077-8</bdi>.</cite></li>

<li><cite id="CITEREFGujaratiPorter2009" class="citation book cs1"><a href="Damodar_N._Gujarati" title="Damodar N. Gujarati">Gujarati, Damodar N.</a>; <a href="Dawn_C._Porter" title="Dawn C. Porter">Porter, Dawn C.</a> (2009). <i>Basic Econometrics</i> (Fifth ed.). New York: McGraw-Hill Irwin. pp. <span class="nowrap">692–</span>698. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-07-337577-9</bdi>.</cite></li>

<li><cite id="CITEREFHayashi2000" class="citation book cs1"><a href="Fumio_Hayashi" title="Fumio Hayashi">Hayashi, Fumio</a> (2000). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=QyIW8WUIyzcC&pg=PA200"><i>Econometrics</i></a>. Princeton University Press. pp. <span class="nowrap">200–</span>203. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-691-01018-8</bdi>.</cite></li>

<li><cite id="CITEREFKmenta1986" class="citation book cs1"><a href="Jan_Kmenta" title="Jan Kmenta">Kmenta, Jan</a> (1986). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Bxq7AAAAIAAJ&pg=PA660"><i>Elements of Econometrics</i></a> (Second ed.). New York: Macmillan. pp. <span class="nowrap">660–</span>672. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-02-365070-2</bdi>.</cite></li>

<li><cite id="CITEREFKoopmans1949" class="citation journal cs1"><a href="Tjalling_Koopmans" title="Tjalling Koopmans">Koopmans, Tjalling C.</a> (1949). "Identification problems in economic model construction". <i><a href="Econometrica" title="Econometrica">Econometrica</a></i>. <b>17</b> (2): <span class="nowrap">125–</span>144. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1905689">10.2307/1905689</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1905689">1905689</a>.</cite> ("A classic and masterful exposition of the subject", <a href="#CITEREFFisher1966">Fisher 1966</a>, p. 31)</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li><cite id="CITEREFLewbel2019" class="citation journal cs1"><a href="Arthur_Lewbel" title="Arthur Lewbel">Lewbel, Arthur</a> (2019-12-01). "The Identification Zoo: Meanings of Identification in Econometrics". <i><a href="Journal_of_Economic_Literature" title="Journal of Economic Literature">Journal of Economic Literature</a></i>. <b>57</b> (4). American Economic Association: <span class="nowrap">835–</span>903. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1257%2Fjel.20181361">10.1257/jel.20181361</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0022-0515">0022-0515</a>.</cite></li>

<li><cite id="CITEREFMatzkin2013" class="citation journal cs1"><a href="Rosa_Matzkin" title="Rosa Matzkin">Matzkin, Rosa L.</a> (2013). "Nonparametric Identification in Structural Economic Models". <i><a href="Annual_Review_of_Economics" title="Annual Review of Economics">Annual Review of Economics</a></i>. <b>5</b> (1): <span class="nowrap">457–</span>486. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1146%2Fannurev-economics-082912-110231">10.1146/annurev-economics-082912-110231</a>.</cite></li>

<li><cite id="CITEREFRothenberg1971" class="citation journal cs1">Rothenberg, Thomas J. (1971). "Identification in Parametric Models". <i>Econometrica</i>. <b>39</b> (3): <span class="nowrap">577–</span>591. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1913267">10.2307/1913267</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0012-9682">0012-9682</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1913267">1913267</a>.</cite></li>

<li><cite id="CITEREFHsiao1983" class="citation cs2">Hsiao, Cheng (1983), <i>Identification</i>, Handbook of Econometrics, Vol. 1, Ch.4, <a href="North-Holland_Publishing_Company" class="mw-redirect" title="North-Holland Publishing Company">North-Holland Publishing Company</a></cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=WlOtUA8Rqw8&index=14&list=PLD15D38DC7AA3B737#t=57m47s"><span class="">Lecture on the identification problem</span></a> on <a href="YouTube_video_(identifier)" class="mw-redirect" title="YouTube video (identifier)">YouTube</a> by <a href="Mark_Thoma" title="Mark Thoma">Mark Thoma</a></li></ul></div><div><div class="zim-footer">

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