Note: 2), 4) and 5) must be for the initial and final year.

8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 98 88 8 8 8 8 8 8 8 8 Module 8 Spatial Econometrics and Regional Income Convergence ECON 760 Economic Analysis. In order to obtain the BLUE property and to able to make statistical inferences about the population parameters (a and b) by means of your estimates (a and b), you need to make certain assumptions about the random part of the regression equation (the random error ) Two of these assumptions are crucial to obtainthe unbiasedness and efficiency of the OLS estimates.\fHypothesis Tests Null hypothesis Ho: a = 0or Ho: b = 0. Alternative hypothesis H1: a 6= 0or H1: b 6= 0. If you reject Ho, the parameter (a or b ) is statistically different fromzero. 100000009090090 90000900000000 AGM UNIVERSITYMust determine if there is sufficient statistical evidence to indicate that Y is truly related to X (i.e., b 6= 0) Even if b = 0, it is possible that the sample will produce an estimate b that is different from zero. Tests for statistical significance . t-test. . p-value.. First determine the level of significance (0.1%, 1%, 5%, 10%) o Probability of finding a parameter estimate to be statistically different fromzero when, in fact, it is zero (alpha). a = 0.001, 0.01, 0.05, or 0.1, respectively. o Probability of a Type | Error (alpha) . 1 - level of significance (alpha) = level of confidence . t-ratio is computed ast = G o where S; is the standard error of estimate b . Use t-table to choose critical t-value with n -k . degrees of freedom for the chosen level of significance on = number of observations o k = number of parameters estimated.Individual Significance-t-Test If the absolute value of t- If t-ratio (in absolute ratio is greater than the value) is equal to 2 (or critical t, the parameter bigger than 2) , you can estimate is statistically reject Ho. significant at the given level of significance.\fJoint Significance -F-test Used to test for significance of overall regression equation Compare F-statistic to critical F-value fromF-table. Two degrees of freedom, n - K & K- 1 Level of significance If F -statistic exceeds the critical F (=4), the regression equation overall is statistically significant at the specified level of significance.Coefficient of Determination: R2 R' measures the percentage of total variation in the dependent variable ( Y ) that is explained by the regression equation Ranges from 0to 1. High R indicates Y and X are highly correlated. E.g. R? = 0.8 means that 80% ofthe changes of Y are explained by the regression equation. 109000009009090 90000900000000 AGM UNIVERSITYSpatial Analysis: Motivation Diagnosis . The assumption of normal, homoskedastic and uncorrelated error terms that lead to BLUE characteristic of OLS estimators are not necessarily satisfied by the real models and data. When dealing with spatial data you must give special attention to the possibility that the errors or the variables (Xs) in the model show spatial dependence.\fSpatial Analysis: Motivation Applied work in regional science (economics, health, demographics, etc.) uses of spatial data. Spatial data: Data collected with reference to location. . Administrative spatial units (states, districts, counties, etc.). . Functional regions (E.g. labour market regions). Points in space (E.g. cities, municipalities, plants). Using spatial data, model estimation, hypothesis testing and prediction have to allow for spatial effects.Spatial Dependence Lack of independence among spatial data, Observations at location / depend on other observations at locations j (6= /). Spatial dependence is associated with the notion of relative space (location) . Neighboring regions are expected to be more alike than arbitrary regions. . Spatial dependence is expected to diminish with increasing distance. Spatial dependence are multidirectional by nature. Time series is unidirectional. AGM UNIVERSITY\fSpatial Dependence: Causes Substantive: Interaction and dependence on the regional level may be itselfa modelling problem because it generats model bias. Location and distance are important forces at workin human geography and market activity. E.g spatial spillovers, hierarchy of places, etc.. This can be corrected byinduding an explicit spatial lag term as an explanatory variable in the model (SAR).Spaal Heterogeneiy .--_.-"'"."'-u E] It refers to 1urarying economic relationships or disturbances over space. '1 "L. A different relationship may hold for every spatial unit. This situation characterizes the case of structural instability. A In case of structural instability, the regression coefficients are not constant across the spatial units. Spatial Heterogeneity E.g., Sample: 35,000 homes sold within the last 5 years in Lucas county, Ohio. . 3 distinct distributions, with low-priced homes nearest to the Central Business District(CBD) and high-priced homes farthest away from the CBD. . This suggests different relationships may be at work to describe home prices in different locations. 090900900900 90000900000000 AGM UNIVERSITY\fW In a regular grid, neighbors can be defined in a number of ways. Among others, you may find: In analogy of the game of chess, rook contiguity, bishop contiguity and queen contiguity are distinguished. Inverse distance raised to a power.\fOLS: Ordinary Least Squares Parameters The coefficients in an equation that determine the exact mathematical relation among the variables (growth rate andinitial income) Unknowns. Parameter estimation The process offinding estimates ofthe numerical values ofthe parameters of an equationX: Spatial units; Y: the weighted average or spatial lag of the corresponding observation on the X axis. 1929 1945 1994 40 They show spatial dependence because there is a positive correlation (See page 146, R & M)Income Convergence Robert Solow (1956)"Capital should flow from countries with a high capital-to-output ratio to countries with a low capital-to- output ratio "Poor countries/regions/states should have higher growth rates. "rich" countries/regions/states should have lower growth rates The analysis using regions is called Regional Income Convergence. 109000009009090 90000900000000 AGM UNIVERSITYOLS: Ordinary Least Squares The purpose of linear regression is to find a (linear) relationship between the dependent variable and a set of explanatory variables. There can be cross-section or times series data. 109090090090900 90000900000000 AGM UNIVERSITYUS states: Exploring -convergence Explore/ Scatter plot/ 1994-29 Y: "d/94/29", X: "LINC29", OK 1945-29 Y: "d/45/29", X: "LINC29", OK 1994-46 Y: 'd/94/46", X: "LINC46", OK You should get negative relationships 00090090009000 90000900000000 AGM UNIVERSITYUS states: Exploring B- convergence 1994-29 1945-29 1994-46 TH X : Initial income & Y : Growth rate. At first glance, B- convergence holds.GeoDa US states - Exploring data The CV of real per capita income in logs across US states falls over time, soo-convergence holds According to Moran's I, data showsspatial dependence. Table: Descriptive Statistics Mean Median SD CV Moran's I LINC29 6.35 6.38 0.38 0.06 0.65 LINC45 7.02 7.03 0.23 0.03 0.57 LINC94 9.96 9.95 0.13 0.01 0.35\fOLS:Bivariate form Y= a+ bX +E Interceptparameter (a ) gives value of Y where regression line crosses Y -axis (value of Y whenX is zero. Slope parameter (b) gives the change in Y associated with a one-unit change in X : AY /AXOLS: Two objectives . Find a good match (fit) between a + bX and observed . values of Y ( a and b are the regression coefficients). b b . Discover which of the explanatory variables (X's) contribute significantly to the linear relationship\fGeoDa : SAR See slide 32 and R & Mpage 150, equation 9 Click on Regression Dependent variable/ growth rate (E.g. d194129) Independent variable/ initial income (E.g. LINC29 ) Weights file/ Spatial lag Three regressions: di94129;= a+ pWd194129;+ BLINC29i+ epi dI45129; = a+ pWdI45129;+ BLINC2gi + epi di941461= a+ pWd194146:+ BLINC461+ 846i Where i : 1, 2, 3, ..., .48GeoDa: SAR -Outcomes: 1994-29 SUMMARY OF CUITPUT: SPATIAL LAG MEAL - MAXIMUM LIKELIHOOD ESTIMATION WINNING Number of observations: 1.6104 Number of Variables O. SHOTN O. ISMFT Nik Info arf marion : O. OGALLON ard. urrar T- VIL O. CHALA CONSTANT O. SOMEA O. CHALCAN -11. 1641 FOR HATHAD ANDASTICLIT RANEM 1. 1241 SPATIAL LAG DEPENDENCE POR KIGHT MATRIX : NUEVA 1 1.7134 O.OHIOL AGM UNIVERSITYGeoDa: SAR-Outcomes: 1945-29 SUMMARY OP OUTPUT: SPATIAL LAG MEAL - MAXIMUM LIKAL CHICO ESTIMATION DINISINT Number of observations: 41 O.67424 Number of variables pages of Brandon O. 30686 Naike Info arf marion -114.075 a. of regression and.unrar DIN SINTO CONSTANT A. TAG -O. HIAM O. OMANALN -A. TIGTA TEST SPATIAL LAG LEPENBACK FOR KICHI MATRIX : NEW TEST AGM UNIVERSITY\fOLS . The OLS regression line (red one) is that minimizes the sum of the squared prediction errors Y = a+bx Sample regression line S = 11.573 + 4.9719A 60 000-46 136- Advertising expenditures (Sallam)