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I have to check if the following models are linear and if so setup the design matrix. In case they are not linear I have to find a transformation $h: \mathbb{R}^n \rightarrow \mathbb{R}^n$ such that $h(Y)$ is linear and then set up the design matrix.

Let $\epsilon_1,..,\epsilon_n$ ~ $\mathcal{N}(0,\sigma^2), n>3$ i.i.d.. Let $x_1,...,x_n$ known and $\beta_1,\beta_2,\beta_3$ unknown real numbers.

  1. $Y_i = \beta_1 + \beta_2 x_i^2 + \epsilon_i$
  2. $Y_i = \beta_1 + \beta_2 x_i + \epsilon_i^2$
  3. $Y_i = e^{\beta_1} e^{\beta_2 x_i} x_i^{\beta_3} e^{\epsilon_i}$

I don't know what part has to be linear. For 1) I'd let the design matrix \begin{matrix} 1 & x_1^2\\ 1 & x_2^2 \\ .... & ... \\ 1 & x_n^2 \end{matrix} Is this correct? Is this model linear? For 2) I'm not sure if its okay that $\epsilon$ is squared. For 3) this doesn't look linear in any way. Is $h = (\log(Y_1),...,\log(Y_n))$ a correct transformation? And design matrix
\begin{matrix} 1 & x_1 & \log{x_1}\\ 1 & x_2 & \log{x_2} \\ .... & ... & ... \\ 1 & x_n & \log{x_n} \end{matrix}

Edit: In the definition of the linear model I have is $X=A\gamma+\sqrt{v}\xi$ and $\xi$ is a standard random vector. So are those only linear models if σ=1?

kjetil b halvorsen
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user391334
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  • If you can write the model as $\hat y = Ax$, then it's linear, for appropriate matrices $\hat y, x, A$ – Firebug Jun 28 '23 at 16:05
  • Your answers are good! concerning the squared errors, you would need to specify their distribution. They have non zero mean, and therefore usual estimation methods would give a biased intercept estimates. The third model is sometimes called log-linear, because it is a linear model if you take the log of Y. – Ute Jun 28 '23 at 16:17
  • You write "I have to". Is that because it is some exercise? In that case could you mark this question as [tag:selfstudy] and follow the suggestion from the link (e.g. explain why you 'have' to do these things, and also explain what you do not understand).... – Sextus Empiricus Jun 28 '23 at 16:20
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    If you can specify the model with a design matrix and an additive error term, then it's linear. – whuber Jun 28 '23 at 17:02
  • You might find https://stats.stackexchange.com/questions/148638 helpful concerning the various definitions of "linear" that can apply here. – whuber Jun 28 '23 at 17:09
  • Thank you all. In the definition of the linear model I have is $X = A \gamma + \sqrt{v} \xi$ and $\xi$ is a standard random vector. So are those only linear models if $\sigma = 1$? – user391334 Jun 28 '23 at 17:36
  • Where is "$\sigma$" in your definition? I suspect you are using "$\sigma$" and "$\sqrt v$" synonymously. In your question (2), is $\epsilon_2$ a standard random vector or not? What is your definition of "standard"? – whuber Jun 28 '23 at 20:44
  • Does this answer your question? How to tell the difference between linear and non-linear regression models? – Sextus Empiricus Jun 29 '23 at 08:13

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