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Suppose I am having a daily unknown variable $y_i$ that can be modeled by $$y_i=\beta_0 + \beta_1x_{1i} +...+ \beta_kx_{ki}$$ where $x_{ji}$ are known for every day $i$.
In order to estimate a model, I have daily $x_{ji}$ plus $Y=\sum_{i=1}^Ny_i$ available. $N$ will be something around 90, so I only observe the endogenous variable on a quarterly basis. Now, if I set $$X_j=\sum_{i=1}^Nx_{ji}$$ I can estimate the parameters of the model $$Y=\beta_0^*+\beta_1X_1+...+\beta_kX_k$$ by using Linear Regression.

The actual aim at this point is to estimate $y_i$ given the model estimated off quarterly observations. To do that I assume that $\beta_0^*=N\beta_0$ and get $$y_i=\frac{i}{N}\beta_0^*+\beta_1x_{1i}+...+\beta_kx_{ki}$$ which will yield reasonable results for $y_i$.

Now my problem is that I have many different systems where I will get less out-of-sample prediction error (I want to minimize prediction error on $y_i$ as much as possible) in the quarterly model when I apply a transformation $\theta$ on both $Y$ and $X_j$: $$\theta(Y)=\tilde{\beta_0^*}+\tilde{\beta_1}\theta(X_{1})+...+\tilde{\beta_k}\theta(X_k)$$ where the transformation I use are $$\theta_1:First\;differences \;of\;quarterly\;observations \;(\Delta_1)$$ $$\theta_2:Percentage\;change\;of\;quarterly\;observations$$ $$\theta_3:Percentage\;change\;of\;quarterly\;observations\;but\;over\;one\;year$$ $$\theta_4:Logarithmic\;transformation$$ $$\theta_5:First\;differences\;of\;Logarithmic\;transformation$$ After transforming I can estimate the new $\tilde{\beta}s$.

Now I will also have to apply $\theta$ on daily $y_i/x_{ji}$. To stay consistent I will transform as if I was transforming the quarterly variables - e.g. if I am working with first differences and did $$\Delta_1Y_t=Y_t-Y_{t-1}$$ I will do $$\Delta_1y_{ti}=y_{ti}-Y_{t-1}$$ and - if I only take the latest quarter $t$ into consideration (parameters were estimated over more than one quarter)- I will get $$y_{ti}-\frac{i}{N}Y_{t-1}=\frac{i}{N}\beta_0^*-\frac{i}{N}\beta_1X_{t-1;1}-...-\frac{i}{N}\beta_kX_{t-1;k}+\beta_1x_{t1i}+...+\beta_kx_{tki}$$ So, by plugging in all the (known) variables on the right-hand side of the equation I will get an estimate $\hat{\tilde{y_{ti}}}=\hat{y_{ti}}-\frac{i}{N}Y_{t-1}$ which I can now simply retransform to get the actual estimate $\hat{y_{ti}}$. (the estimates were sanity checked and I assumed that everything I did up to this point was correct, but feel free to correct me) For $\theta_2$ (percentage change) I came up with: $$\frac{y_{ti}}{Y_{t-1}}-\frac{i}{N}=\frac{i}{N}\beta_0^*-\frac{i}{N}\beta_1-...-\frac{i}{N}\beta_k+\beta_1\frac{x_{t1i}}{X_{t-1;1}}+...+\beta_k\frac{x_{tki}}{X_{t-1;k}}$$ but this seems to give me wrong estimates which at least have a linear relation to what I assume to be reasonable estimates when I plot them.

So, my question is: How do the proper models for $\theta_{2-5}$ look like? (if that's too much for you, I would be happy with $\theta_2$ :) )

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