Is log-linear regression a generalized linear model? Does log-linear regression fall into the class of generalized linear models? Here I'm defining "log-linear regression" as the model $\log(y) = x'\beta + \eta$ where $\eta \sim N(0, \sigma^2)$.
Thanks.
 A: Normally, loglinear models for contingency tables are considered as generalized linear models (Fox 2016). They are sometimes called Poisson regression for contingency tables (Bilder & Loughlin 2015). In the case of Poisson regression, we have a response random variable $Y$, and $p \geq 1$ explanatory variables, $x_1,\dots,x_p$, and for observations $i=1,\ldots,n$, we assume that 
$$Y_i \sim Po(\mu_i)$$
where 
$$\mu_i = \exp(\beta_0+\beta_1x_{i1}+\ldots+\beta_px_{ip}).$$
So, the generalized linear model has a Poisson random component, linear predictor (systematic component), and link function (log-link):
$$\log(\mu)=\beta_0+\beta_1x_1+\ldots+\beta_px_p.$$
Looks similar to your equation. However, first, you seem to transform the outcome, $y$, directly. So, it looks like you follow what Agresti (2015:6) calls transformed-data approach (i.e., $E[g(y_i)] = \beta_0+\beta_ix_{i1}$ instead of $g[E(y_i)]= \beta_0+\beta_ix_{i1}$, $g$ is the link function). And second, (I think) you specify error distribution, $\eta \sim N(0, \sigma^2)$. As you can see in this answer, in GLMs, "You don't specify the "error" distribution, you specify the conditional distribution of the response." The exception is latent variable approach.
To answer your question: yes, log-linear regression falls into the class of generalized linear models, but your model looks like a linear regression model with a log-transformed outcome. 

Agresti, Alan. 2015. Foundations of Linear and Generalized Linear Models. Hoboken, NJ: Wiley.
Bilder, Christopher R. and Thomas M. Loughlin. 2015. Analysis of Categorical Data with R. Boca Raton, London and New York: CRC Press.
Fox, John. 2016. Applied Regression Analysis and Generalized Linear Models. 3rd ed. Los Angeles: Sage Publications.
