Determine whether the equation is linear in the variables x and y, s.pdf

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Determine whether the equation is linear in the variables x and y, show work and explain... (sin 2)x-y=14 Solution Yes, given equation is linear in both x and y. For an equation to be linear in some variable, the given variable should have highest power 1. Here, this condition is satisfied as both x and y have highest power 1. Coefficient of these variables should be real numbers. You can see that this condition is also satisfied as x has coefficient sin(2) which is a real number, y has coefficient of -1 which is also a real number. Note that for an equation to be linear, there should not be a term involving multiple of two or more variables. For example, if there was a term let\'s say xy, then it was going to be non-linear equation as variables x and y are being multipled by each other..

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2. If you have a nonlinear relationship between an independent variable x and a dependent variable y, how can you apply the least -squares fit method? Does this work for all nonlinear relationships? if not please give an example of a nonlinear equation where the fit method cannot be used.(No need to use Matlab on this question). Solution A)The assumptions of linearity and additivity are both implicit in this specification. • Additivity = assumption that for each IV X, the amount of change in E(Y) associated with a unit increase in X (holding all other variables constant) is the same regardless of the values of the other IVs in the model. That is, the effect of X1 does not depend on X2; increasing X1 from 10 to 11 will have the same effect regardless of whether X2 = 0 or X2 = 1. • With non-additivity, the effect of X on Y depends on the value of a third variable, e.g. gender. As we’ve just discussed, we use models with multiplicative interaction effects when relationships are non-additive. Linearity = assumption that for each IV, the amount of change in the mean value of Y associated with a unit increase in the IV, holding all other variables constant, is the same regardless of the level of X, e.g. increasing X from 10 to 11 will produce the same amount of increase in E(Y) as increasing X from 20 to 21. Put another way, the effect of a 1 unit increase in X does not depend on the value of X. • With nonlinearity, the effect of X on Y depends on the value of X; in effect, X somehow interacts with itself. This is sometimes refered to as a self interaction. The interaction may be multiplicative but it can take on other forms as well, e.g. you may need to take logs of variables. Dealing with Nonlinearity in variables. We will see that many nonlinear specifications can be converted to linear form by performing transformations on the variables in the model. For example, if Y is related to X by the equation E Yi Xi ( ) = + 2 and the relationship between the variables is therefore nonlinear, we can define a new variable Z = X2 . The new variable Z is then linearly related to Y, and OLS regression can be used to estimate the coefficients of the model. There are numerous other cases where, given appropriate transformations of the variables, nonlinear relationships can be converted into models for which coefficients can be estimated using OLS. We’ll cover a few of the most important and common ones here, but there are many others. Detecting nonlinearity and nonadditivity. The key question is whether the slope of the relationship between an IV and a DV can be expected to vary depending on the context. • The first step in detecting nonlinearity or nonadditivity is theoretical rather than technical. Once the nature of the expected relationship is understood well enough to make a rough graph of it, the technical work should begin. Hence, ask such questions as, can the slope of the relationship between Xi and E(Y) be expected to have the same sign for all value.

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