Is positive coefficient of price correct in a multiple regression model I am currently undertaking forecasting of energy sales (kWh) for our industrial customers. From historical data gathered from 1993 to 2013, a graph of price per kwh against sales kwh shows a positive relationship. Going ahead to generate the model yielded a positive coefficient of price with energy sales. The other independent variables are GDP of the industrial sector and the lag of sales for 1, 4 and 6. What can explain the positive coefficient of price with sales? Plotting GDP of the industrial sector yields a positive relationship with sales (kWh). Can someone help? I need to understand why we have a positive slope of price with sales. The historical data clearly confirms that.
 A: Picking up on Dimitriy V. Masterov's comment, a demand curve sloping upwards usually means that you have simultaneous equations bias.  Here is the equation you are trying to estimate:
\begin{align}
D: Q_t &= \beta_1 + \beta_2P_t + \epsilon_t
\end{align}
As long as $Cov(P,\epsilon)=0$, you can estimate this equation using ordinary least squares regression with no problem.  In this case, you expect to get an estimate of $\beta_2$ which is negative.  However, if $Cov(P,\epsilon)>0$, then you can expect the OLS estimate of $\beta_2$ to be too high.  If that positive correlation is strong enough, then you can get a positive estimate for $\beta_2$ even when the true value of $\beta_2$ is negative.
How can $Cov(P,\epsilon)>0$ arise in practice?  There are lots of ways.  Suppose that your company prices electricity higher in time periods of high demand.  In the equation, time periods of high demand are time periods when $\epsilon$ is higher than average.  So, in this case, $P$ will be above its average when $\epsilon$ is above its average---more or less the definition of $Cov(P,\epsilon)>0$.  Another possibility is that your company prices electricity using some formula like "cost of fuel plus 5%."  Then, if the cost of fuel is high when the economy is booming and also the demand for electricity is high when the economy is booming, then you will get $Cov(P,\epsilon)>0$ through that mechanism.  So, the first thing it will be useful for you to do is to find out how your company prices electricity, find out what non-price factors affect demand for electricity from your industrial customers, and then think about how those things are correlated.
Sometimes, you can more or less fix the problem just by including control variables.  Suppose you find out that the story I told above about the overall economy booming or busting is the key driver.  Then, you may be able to fix the problem just by including GDP on the right-hand-side of the equation.
Even better, though, is to find something called an instrumental variable.  An instrumental variable would be a variable 1) which is correlated with $P$, 2) which does not belong in the equation, and 3) which is also not correlated with $\epsilon$.  In this case, you want some variable which helps to determine electricity price but which is not correlated with unobserved demand.  Usually, in electricity generation, you are going to use the price of your key fuel as an instrument.  Or, if you use several fuels, an appropriately weighted average.  The trick is that you have to include the right variables on the right-hand-side to ensure that the correlation between $\epsilon$ and the instrument is zero.  So, to defeat my story about boom times also being times when fuel stocks are expensive (this would make fuel price a bad instrument) you would need to include GDP as a right hand side variable.
What do you do with the instrumental variable?  Well, you plug it into an estimation technique called two-stage least squares.
Shorter answer:  you have endogeneity/simultaneous equations bias; use instrumental variables.
