# hotelling model example

After all, we also think that the price of oil is determined by demand and supply in a market. Each consumer will purchase one unit of good from the rm that is closest to him. the location of different sellers in a market respect to one another. Movies, too, are designed to be enjoyed by particular ages. Example 7.37: calculation of Hotelling's T^2 Hotelling's T^2 is a multivariate statistic used to compare two groups, where multiple outcomes are observed for each subject. For example, movie customers are differentiated by age, and we can represent moviegoers by their ages. 3.1. Applied Multivariate Statistical Analysis, 7.1.4 - Example: Women’s Survey Data and Associated Confidence Intervals, Lesson 1: Measures of Central Tendency, Dispersion and Association, Lesson 2: Linear Combinations of Random Variables, Lesson 3: Graphical Display of Multivariate Data, Lesson 4: Multivariate Normal Distribution, 4.3 - Exponent of Multivariate Normal Distribution, 4.4 - Multivariate Normality and Outliers, 4.6 - Geometry of the Multivariate Normal Distribution, 4.7 - Example: Wechsler Adult Intelligence Scale, Lesson 5: Sample Mean Vector and Sample Correlation and Related Inference Problems, 5.2 - Interval Estimate of Population Mean, Lesson 6: Multivariate Conditional Distribution and Partial Correlation, 6.2 - Example: Wechsler Adult Intelligence Scale, Lesson 7: Inferences Regarding Multivariate Population Mean, 7.1.1 - An Application of One-Sample Hotelling’s T-Square, 7.1.8 - Multivariate Paired Hotelling's T-Square, 7.1.11 - Question 2: Matching Perceptions, 7.1.15 - The Two-Sample Hotelling's T-Square Test Statistic, 7.2.1 - Profile Analysis for One Sample Hotelling's T-Square, 7.2.2 - Upon Which Variable do the Swiss Bank Notes Differ? First introduced in a paper by Harold Hotelling in 1929, the model still holds today. complements in the Hotelling model. Now we are ready to define the Two-sample Hotelling's T-Square test statistic. Suppose there are two gas stations, one located at 1 4 and the other located at 1. Hotelling's theory addresses a fundamental decision for an owner of a non-renewable resource: keep the resource in the ground and hope for a better price the next year, or extract and sell it … The example of firms locating on a street is boring, so I show the student’s Wired’s classic “Battle for Blue.” I think I’ll start using this one now (via Scott Rouse). $$T^2 = \mathbf{(\bar{x}_1 - \bar{x}_2)}^T\{\mathbf{S}_p(\frac{1}{n_1}+\frac{1}{n_2})\}^{-1} \mathbf{(\bar{x}_1 - \bar{x}_2)}$$. The efficient use of scarce natural resources, both renewable and non-renewable sources, has long been a concern of natural resource economics (Shogren 2000). Suppose further that there are 100 customers located at even intervals along this beach, and that a customer will buy only from the closest vendor. Hotelling’s T²¶. So, as before, we will look at transforming this Hotelling's T-square statistic into an F-statistic using the following expression. For easy comprehension, the Hotelling model is sometimes explained by using the example of a beach where two ice cream stands are trying to decide their best locations. If two or more rms are located at equal distance, consumer will buy from each rm with equal probability. A more preferable test statistic is Hotelling’s $$T^2$$ and we will focus on this test. The same line is used to represent products. The unidirectional Hotelling model where consumers can buy only from firms located on their right (left) is extended to allow for elastic demand functions. Hotelling’s theory is a simple and straightforward concept. For small samples, the chi-square approximation for $$T^2$$ does not take into account variation due to estimating $$Σ$$ with the sample variance-covariance matrix S. Better results can be obtained from the transformation of the Hotelling $$T^2$$ statistic as below: $F = \frac{n-p}{p(n-1)}T^2 \sim F_{p,n-p}$. The final quantity from a PCA model that we need to consider is called Hotelling’s $$T^2$$ value. Trained in mathematics, he participated in the early twentieth century movement to mathematize economics. The sample variance-covariance for the second sample of notes, the counterfeit note, is given below: $$S_2 = \left(\begin{array}{rrrrrr}0.124&0.032&0.024&-0.101&0.019&0.012\\0.032&0.065&0.047&-0.024&-0.012&-0.005\\0.024&0.047&0.089&-0.019&0.000&0.034\\-0.101&-0.024&-0.019&1.281&-0.490&0.238\\ 0.019&-0.012&0.000&-0.490&0.404&-0.022\\0.012&-0.005&0.034&0.238&-0.022&0.311\end{array}\right)$$. However, as before this approximation does not take into account the variation due to estimating the variance-covariance matrix. Examples. A Bertrand-type model … Note that this database has a fourth variable, Disability, that is ignored in this analysis. Yet, he’s perhaps best known for some simple yet profound observations, one of which is now known as Hotelling’s Law. After concluding that the counterfeit notes can be distinguished from the genuine notes the next step in our analysis is to determine upon which variables they are different. Examples of location models include Hotelling’s Location Model, Salop’s Circle Model, and hybrid variations. It has spawned numerous papers on the extrapolation of its concepts. Hotelling's Model. Linear Hotelling model 1 Town with just one street of length 1, along which all reside. Two pizza places located at a and 1 b. In the expression for Hotelling's $$T^2$$, the difference between the sample mean and $$\mu_{0}$$ is replaced with the difference between the sample mean vector and the hypothesized mean vector $$\boldsymbol{\mu _{0}}$$. Some PCA models will have many components, $$A$$, so an initial screening of these components using score scatterplots will require reviewing $$A(A-1)/2$$ scatterplots. Treatment is the group variable. Hotelling[{0,.6,1},0,10,100] solves the Hotelling model with initial product positions at 0,.6 and 1, no entrant, homogenous marginal costs of 10 and a market size of 100. (ii) The distribution of customers is uniform on the segment (with unit There are N= 4 rms. A good short video to use when teaching or learning about game theory. Example 7.37: calculation of Hotelling's T^2. Now consider squaring this test statistic as shown below: $t^2 = \frac{(\bar{x}-\mu_0)^2}{s^2/n} = n(\bar{x}-\mu_0)\left(\frac{1}{s^2}\right)(\bar{x}-\mu_0) \sim F_{1, n-1}$. THE MODEL The assumptions of the standard 3-firm Hotelling location model are as follows: (i) Three firms i = 1, 2,3 locate on a segment of unit length, at locations xi (i = 1, 2,3) and sell a homogeneous commodity. The $$T^2$$ value for the $$i^\text{th}$$ observation is defined as: T… Hotelling's Model. (In the Hotelling model, this comes for example from using the square of the distance a voter must travel.) Consider the last term in the above expression for $$t^2$$. II. The best response curves intersect at the equilibrium prices pN 1 = pN 2 = 12 as shown below, leading to proﬁts of π1 (12,12) = π2 (12,12) = 144. MODEL EXAMPLE: Hotelling’s Model of Spatial Aggregation: Suppose the buyers of a commodity are uniformly distributed along a line segment of length l. Two vendors A and B are at distances a and b, respectively, from each end of the line segment: o The cost of production for the good to A and B is assumed to be zero. Initially, there are slightly under 30 billion barrels of oil available. The existence of endpoints in the space of characteristics produces results which are qualitatively different from the ones of the circumference model. The Hotelling model is a commonly adopted approach when buyers’ preferences are considered in a duopoly market. This is a function of the sample sizes of the two populations and the number of variables measured p. $$F = \dfrac{n_1+n_2-p-1}{p(n_1+n_2-2)}T^2 \sim F_{p, n_1+n_2-p-1}$$. This database is stored in the file MANOVA1. In the Hotelling model, customers' preferences are located by points on the same line segment. Hotelling’s model is actually a good argument for protectionist policies. Excepturi aliquam in iure, repellat, fugiat illum voluptate repellendus blanditiis veritatis ducimus ad ipsa quisquam, commodi vel necessitatibus, harum quos a dignissimos. Suppose there are two gas stations, one located at 1 4 and the other located at … Test for Relationship Between Canonical Variate Pairs, 13.4 - Obtain Estimates of Canonical Correlation, 14.2 - Measures of Association for Continuous Variables, 14.3 - Measures of Association for Binary Variables, 14.4 - Agglomerative Hierarchical Clustering, Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris, Duis aute irure dolor in reprehenderit in voluptate, Excepteur sint occaecat cupidatat non proident. For large samples, this test statistic will be approximately chi-square distributed with $$p$$ degrees of freedom. Although the positive results of content mixing in the Hotelling model coincide with the ones of the standard analysis, the normative economics are different. The sample mean vectors are copied into the table below: The sample variance-covariance matrix for the real or genuine notes appears below: $$S_1 = \left(\begin{array}{rrrrrr}0.150& 0.058& 0.057 &0.057&0.014&0.005\\0.058&0.133&0.086&0.057&0.049&-0.043\\0.057&0.086&0.126&0.058&0.031&-0.024\\0.057&0.057&0.058&0.413&-0.263&-0.000\\0.014&0.049&0.031&-0.263&0.421&-0.075\\0.005&-0.043&-0.024&-0.000&-0.075&0.200\end{array}\right)$$. Hotelling Model The transportation costs of consumer x: Of buying from seller A are Of buying from seller B are s ≡gross consumer surplus - (i.e. We reject $$H_{0}$$ at level $$α$$ if $$t^2$$ is greater than the critical value from the F-table with 1 and n-1 degrees of freedom, evaluated at level $$α$$. Suppose that two owners of refreshment stands, George and Henry, are trying to decide where to locate along a stretch of beach. Breakfast cereals are primarily distinguished by their sugar content, which ranges on the Hotelling line from low on the left to high on the right. Example 1: The shoe company from Example 1 of One Sample Hotelling’s T 2 Statistic is considering phasing out an existing shoe model (Model 2) by the prototype described in Example 1 of One Sample Hotelling’s T 2 Statistic. Hotelling’s Law is also referred to as the principle of minimum differentiation or Hotelling’s linear city model. Breakfast cereals are primarily distinguished by their sugar content, which ranges on the Hotelling line from low on the left to high on the right. 0 2 4 6 8 10 12 14 16 16 14 12 10 8 6 4 2 0 p1 p2 Hotelling Best Responses 2JointProﬁt Maximization We reject the null hypothesis, $$H_{0}$$, at level $$α$$ if the test statistic F is greater than the critical value from the F-table with p and n-p degrees of freedom, evaluated at level $$α$$. Anthony Downs saw that this model could explain some aspects of political competition of candidates with respect to ideological position. model on [O, 11 of Hotelling (1929) and the oligopoly circumference model of Salop (1979). Hotelling Model. Hotelling's T^2 is a multivariate statistic used to compare two groups, where multiple outcomes are observed for each subject. For example, as shown by Anderson and Neven (1989), the socially optimal locations in the mixing model are indeed 0 and 1. Examples of location models include Hotelling’s Location Model, Salop’s Circle Model, and hybrid variations. At the top of the first output page you see that N1 is equal to 100 indicating that we have 100 bank notes in the first sample. It clearly illustrates the trend that prices of exhaustible resources are likely to take in the future. The counterfeit notes can be distinguished from the genuine notes on at least one of the measurements. Applied Multivariate Statistical Analysis, 7.1.15 - The Two-Sample Hotelling's T-Square Test Statistic, Lesson 1: Measures of Central Tendency, Dispersion and Association, Lesson 2: Linear Combinations of Random Variables, Lesson 3: Graphical Display of Multivariate Data, Lesson 4: Multivariate Normal Distribution, 4.3 - Exponent of Multivariate Normal Distribution, 4.4 - Multivariate Normality and Outliers, 4.6 - Geometry of the Multivariate Normal Distribution, 4.7 - Example: Wechsler Adult Intelligence Scale, Lesson 5: Sample Mean Vector and Sample Correlation and Related Inference Problems, 5.2 - Interval Estimate of Population Mean, Lesson 6: Multivariate Conditional Distribution and Partial Correlation, 6.2 - Example: Wechsler Adult Intelligence Scale, Lesson 7: Inferences Regarding Multivariate Population Mean, 7.1.1 - An Application of One-Sample Hotelling’s T-Square, 7.1.4 - Example: Women’s Survey Data and Associated Confidence Intervals, 7.1.8 - Multivariate Paired Hotelling's T-Square, 7.1.11 - Question 2: Matching Perceptions, 7.2.1 - Profile Analysis for One Sample Hotelling's T-Square, 7.2.2 - Upon Which Variable do the Swiss Bank Notes Differ? The two sample Hotelling's $$T^{2}$$ test can be carried out using the Swiss Bank Notes data using the SAS program as shown below: Download the SAS Program: swiss10.sas. The inverse of the sample variance is replaced by the inverse of the sample variance-covariance matrix S, yielding the expression below: $$T^2 = n\mathbf{(\overline{X}-\mu_0)'S^{-1}(\overline{X}-\mu_0)}$$. It also involves a calculation of the pooled variance-covariance matrix multiplied by the sum of the inverses of the sample size. (Hotelling, 50) In another example, Hotelling noted that if instead of determining prices endogenously they were fixed, such as in a socialist regime, firms now locate themselves at the center of the space. Suppose there are two competing shops located along the length of a street running north and south, with customers spread equally along the street. For large n, $$T^2$$ is approximately chi-square distributed with p degrees of freedom. View the video below to see how to compute the Two Sample Hotelling's $$T^2$$ using the Minitab statistical software application. The unidirectional Hotelling model where consumers can buy only from firms located on their right (left) is extended to allow for elastic demand functions. Robert Malthus raised concern a… For a better comprehension, Hotelling’s model is sometimes explained by using the example of a beach were two ice cream stands are trying to decide the best location. Under null hypothesis, $$H_{0}\colon \boldsymbol{\mu} = \boldsymbol{\mu_{0}}$$, this will have a F distribution with p and n-p degrees of freedom. Each customer will always choose the nearer shop as it is disadvantageous to travel to the farther. Under the null hypothesis, $$H_{o}\colon \mu_{1} = \mu_{2}$$ this F-statistic will be F-distributed with p and $$n_{1} + n_{2} - p$$ degrees of freedom. The duopoly market is a simpliﬁed scenario and can be referred to when analyzing a more complicated oligopoly market. Previous empirical studies of the Hotelling model, though relatively few, generally ﬁnd the theory a poor depiction of the real world. As two competitive cousins vie for ice-cream-selling domination on one small beach, discover how game … This seems a little bit mysterious. Recall that under the null hypothesis t has a distribution with n-1 degrees of freedom. • Click on the file T2.NCSS. Some PCA models will have many components, $$A$$, so an initial screening of these components using score scatterplots will require reviewing $$A(A-1)/2$$ scatterplots. The harder it is for a Republican to shift away from positions needed to win a primary, the easier it is for the Democrats. Women’s Health Survey: One-Sample Hotelling's T-Square Section In 1985, the USDA commissioned a study of women’s nutrition. Harold Hotelling was an accomplished economist. This paper presents an evaluation or analysis of Harold Hotelling’s theory that asserts that the most socially and economically profitable extraction track of a non-renewable resource is one along which the price of the resource, determined by the marginal net revenue from the sale of the resource, increases at the rate of interest The paper presents a model of the Hotelling rule and examines its applicability to real life phenomena. (In the Hotelling model, this comes for example from using the square of the distance a voter must travel.) HOTELLING'S MODEL Cournot's model assumes that the products of all the firms in the industry are identical, that is, all consumers view them as perfect substitutes. Test is exactly chi-square distributed with p degrees of freedom now we are ready to define the Two-sample Hotelling \... Holds today can represent moviegoers by their ages will focus on this test Depletion According to Hotelling. 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