## 1 Introduction  
The benefits of free markets derive from their \"efficiency\", i.e., the idea that efficient markets accurately and rapidly impound new information into prices via trading. Market efficiency implies no information remains to be impounded in the price, hence, all new information is a surprise and is inherently unpredictable. Price movements in efficient markets are random, and consequently, conditioning on current and past information, reliance on prediction algorithms is futile. Our goal in this paper is to extend extant tests of market efficiency based on simple statistics to models based on deep learning neural networks. There is a long history of research on market efficiency, beginning with [1]. In a follow up article more than three decades later, [PERSON] [2] revisited the evidence on market efficiency, and found in favor of its broad existence using effective methods, especially event studies. These ideas have existed in the finance literature for a few decades. More recently, there has been a machine-learning-based literature [3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13] that has also explored market prediction with some success, though the jury is still out on whether markets are inefficient and easy to beat. Tests of market efficiency take many forms, such as tests of autocorrelation on univariate time series stock (or stock index) return data, as in [14]. These tests of the weak-form market efficiency hypothesis take a narrow view of the information set on which the tests are conditioned, i.e., the history of the time series being predicted. Event study tests, beginning with [15] also use information from the same time series, and complement this data with the return on the market index. These tests in general, also seem to support overall market efficiency, though of course, some anomalies do exist in the short-term. Tied up with an assessment of affirmative market efficiency is a corresponding proposition of an absence of stock predictability. In this paper, we revisit whether markets are predictable, in particular, if the direction of the broad market, not just one stock, might be predicted with higher than random chance. Our approach, unlike existing methodologies, considers all stocks of the index and employs deep learning schemes to investigate market unpredictability. Two clear differences with existing approaches may be highlighted. First, whereas weak-form tests have relied upon the history of a single time series of returns, we will assess the predictability of the daily direction of S&P 500 index using historical data on _all_ stocks in the index for a preceding period. Therefore, this is a large-scale generalization of the information set used in testing market efficiency. More complex techniques other than standard statistical ones have also been used where predictability surrounding turbulent market events is detected, see [7]. Such approaches still use single time series for prediction and may not work well in normal periods. Second, models for market efficiency tests have usually been based on linear statistical specifications, such as multivariate regression. Here, we will use highly nonlinear deep neural networks to specify the functional relation between the sign of the move in the S&P 500 on day \((t+1)\) and the \(T\)-day history of all stocks in the index until day \(t\). The hope is that what linear models cannot pick up, we may train nonlinear models to learn. Furthermore, by expanding the information set by a huge order of magnitude, we are able to re-examine weak form efficiency more comprehensively. Using the expanded information set, we find that deep learning approaches, with their nonlinear structures do better than a panoply of other comparison prediction algorithms, though not well enough to reject a finding of market efficiency. Market efficiency, evidence by a lack of predictability is usually supported by two features of the data. First, _randomness_. A purely random series is, by definition, unpredictable. Second, _nonstationarity_.