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Simple Linear Regression Analysis Model
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Added on 2020-05-16
Simple Linear Regression Analysis Model
Added on 2020-05-16
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Question 1:1.1 Consider the following simple regression equation y=β0 + βx + ε,where, x, y and ε are n×1 vectors, β0 and β are scalars.To prove : ˆSxySxxThis has been proved using OLSE which fix a data in such a way where Residual Sum ofSquares (RSS) is minimum. RSS=2220111ˆˆˆ()()nnniiiiiiiieyyyx0ˆˆ()ˆiiiiiieyxeyyIn order to find the value of β0 and ˆ the value of 21niie has to be minimum, for which firstorder derivation of RSS with respect to β0 and ˆ has to be performed. The derivation is asfollows:00,0ˆˆRSSRSSandThis leads to,2010ˆˆ((()))0ˆniiiyx, and 201ˆˆ((()))0ˆniiiyxTaking derivative of this further results into0011ˆˆˆˆ2() 0,2() 0nniiiiiiiyxandxyx
From these equations, two equations occurs such as:01ˆˆ() 0niiiyx ...............(1)01ˆˆ() 0niiiixyx .................(2)Here, equation (1) and (2) are normal equations. Solving to these two equation will give β0 andˆ values.Equation (1) can be calculated further as shown below:0111011ˆˆ0ˆˆ0nnniiiiinniiiiyxynx0111100ˆˆˆˆˆˆnniiiinniiiinyxyxnnyxThis calculated value of β0 has been further used to calculate the value of ˆ in equation (2).
011111111111ˆˆ() 0ˆˆ() 0ˆ()() 0ˆ()() 0ˆ()()()ˆ()()()ˆ()()ˆniiiiniiiiniiiinniiiiiinniiiiiiniiiniiiniiiniiixyxxyyxxxyyxxxyyxxxxxxxyyxyyxxxxxyyxxxxSxyxxzSSThus the value of ˆ has been achieved as desired.1.2 Let rxy be the correlation between x and y. Express the relationship between rxy and βˆ.Explain in which case they are equal.Answer:In order to find the relationship of correlation and ˆ the following equation has been used.
10111121( , )() 0()()()()(())()()()()niiiniiiiniiiiniiiiniiyixynxiiCorrexexxyxxxyyxxxxyyxxxxxxyySrSxx where, 0ˆˆyxWhere, ySis standard deviation for y, and, xSis standard deviation for x.From the equation yxyxSrS it can be said that the value of ˆ would be equal to xyr, if xi and yiare such as yS=xS. 1.3 Write down an expression for cos θ, where θ is an angle between x and y, using the lawof cosines. Explain each step clearly. Compare your result with rxy. Interpret your findings The cosine rule which is used to show angle between twon-dimensional vectors xand yis asshown below,22,cosxyxyThe measure of distance from the origin can be presented using an n-dimensional vector which is122,31(,,.......,)ntnttXXXXisX.Following the similar way 2((( )) )EXEX is a measure of the magnitude by which a randomvariable deviates from its mean. ......Consider this as (3)
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